text:
\section{Introduction}

A  $p \times p$  matrix $\mu = (\mu_{i,j})^p_{i,j=1}$ of signed real  measures $\mu_{i,j}$, such that for each Borel set $A \subset [0,1]$ the matrix $\mu(A) = (\mu_{i,j}(A))^p_{i,j=1}$ is symmetric and nonnegative definite, is called matrix measure  on the interval $[0,1]$.
Matrix measures have been studied extensively in the literature generalising many classical results  in the context of  moment theory, orthogonal
polynomials, quadrature formulas [see  \cite{krein1949} for an early reference and   \cite{durass1995},
 \cite{duran1999} and \cite{durlop1997}, \cite{gruenbaum2003}, \cite{grupactir2005},
\cite{dampussim2008}, \cite{gamnag2012} and \cite{gamnag2016}  for some more recent references among many others].  \\
In a recent paper \cite{detnag2012a}  defined  $ \mathcal{P}_p$ as the set of all matrix measures on the interval $[0,1]$
satisfying the condition  $ \int^1_0 d\mu(x) =I_p $ [here and throughout this paper $I_p$ denotes the $p \times p$  identity matrix]
and  studied a uniform distribution on the  moment space
\begin{eqnarray} \label{1.1a}
\mathcal{M}_n^p= \Big \{ (M_1,\dots,M_n)^t \  \Big | \ M_j = \int^1_0 x^j d\mu (x), \ j=1,\dots,n ~,~\mu  \in \mathcal{P}_p \Big \}
\end{eqnarray}
of all (matrix) moments
\begin{eqnarray} \label{1.1}
M_k = \int^1_0 x^k d\mu(x) =  \Big(\int_0^1  x^k d \mu_{i, j}(x)\Big )_{1 \le i, j \le p} ~;\quad k=0,1,2,\dots
\end{eqnarray}
up to the order $n$ corresponding to matrix  measures on the interval $[0,1]$. Note that $ \mathcal{M}_n^p  \subset
(\mathcal{S}_p)^n $, where $\mathcal{S}_p$ denotes the set of all nonnegative definite matrices of size $p \times p$.
If $(M_{1,n}^p, \dots, M_{n,n}^p)^t$ is a uniformly distributed vector on $\mathcal{M}_n^p$,
\cite{detnag2012a}  showed that an appropriately centered and standardized version of the vector $(M_{1,n}^p, \dots, M_{k,n}^p)^t$ converges weakly
to a  vector of $k$ independent $p \times p$ Gaussian ensembles, thus generalising the meanwhile classical results of \cite{chakemstu1993}
for the case $p=1$.
 \\
The one-dimensional case has also been studied intensively with respect to other properties of random moment sequences and we refer to
\cite{gamloz2004}, \cite{lozada2005} for large deviation results and to \cite{detnag2012b} for some results on more general moment spaces.
Recently \cite{dettom2016} examined the asymptotic properties  of a stochastic process of  Hankel determinants
$$
\big \{  \log  \det (M_{i+j,2n}^p) _{i,j=0,\ldots , \lfloor nt  \rfloor}  \big   \} _{t \in [0,1]}
$$
of a uniformly distributed moment vector $(M_{1,2n}^p, \ldots , M_{2n,2n}^p )^t$ on the moment space  $\mathcal{M}_{2n}^p   $ in the case $p=1$ and derived weak convergence and large deviation principles for this process.

In the present paper we will investigate properties of  a stochastic process corresponding to the determinant of  matrix valued random Hankel matrices, where
the dimension of the moment space and the dimension $p_n$ of the matrix measures converge to infinity. To be
precise, consider a uniformly random vector $(M_{1, 2n + 1}^{p_n} , \ldots , M_{2n+1, 2n + 1}^{p_n })^t$ on the moment space $\mathcal{M}_{2n+1}^{p_n}$, where $p_n$ is a sequence of integers converging
to infinity as $n\to \infty$ and define the stochastic process
\begin{equation} \label{basicproc}
{H}_{n} (t) =
\big \{  \log  \det (M_{i+j, 2n + 1} ^{p_n} ) _{i,j=0,\ldots , \lfloor nt  \rfloor}  \big   \} _{ t \in [0,1]}.
\end{equation}
We establish weak convergence of the process  $ \{{H}_{n} (t)   \} _{t \in [0,1]} $ with a Gaussian limit, Mod-Gaussian convergence (for fixed $ t \in [0,1]$)
and derive several moderate and large  deviation principles. In Section~\ref{section_basics} we will present
 some basic facts about matrix-valued moment spaces.
Section~\ref{section_distribution} is devoted to the investigation of  distributional properties of  determinants corresponding to subblocks
of  the Jacobi-beta-ensemble.  These results are of own interest and provide a new point on classical
results about a Bartlett- type decomposition [see  \cite{bart1933}]  for the Jacobi-beta-ensemble [see  for example \cite{kshi1961}]. In Section
\ref{section_applied_convergence} we prove weak convergence of the  process \eqref{basicproc}.
 Finally, in Section~\ref{section_mod_phi} we  examine mod-$\phi$-convergence, as well as moderate and large deviations.
 Our results are based on several delicate estimates  of the cumulants of logarithms of beta-distributed random variables, stated in the
  Appendix, which also contains proofs of the more technical results and  some  inequalities about polygamma functions.

\section{Moment spaces of  matrix-valued meausres}
\label{section_basics}

  We begin recalling some basic facts about the moment space
       $\mathcal{M}_n^p$    defined in \eqref{1.1a}  [see   \cite{detstud2002}  for a detailed discussion].
   In the following we compare matrices   with respect to the Löwner (partial) ordering. Thus for two $p\times p$ symmetric matrices $A, B $
  we use the notation $   A < B$  ($ A \le B $) if and only if the difference $B - A$  is positive definite (positive semi-definite).
We denote by $\mathcal{S}_p$ the set of nonnegative definite (symmetric) $p \times p$ matrices. Let $\mathcal{B}([0,1])$ denote the Borel field on  the interval $[0,1]$.
A map
$$
\mu = (\mu_{i, j})_{1 \le i, j \le p}: \mathcal{B}([0,1]) \to \mathbb{R}^{p \times p}
$$
is called a \emph{matrix-valued measure}, if $\mu_{i, j}$ is a signed measure for all $1 \le i, j \le p$ and $\mu(A) \in \mathcal{S}_p$ for every Borel set $A \subset  [0,1]$.  We denote by $\mathcal{P}_p$ the set of all $p \times p$ matrix-valued
measures on  the interval $[0,1]$ satisfying  $\mu([0,1]) = I_p$ and consider the $n$th moment space $\mathcal{M}_n^p$ defined in \eqref{1.1a},
    which is a subset of $(\mathcal{S}_p) ^n$. Note that in the one dimensional case $\mathcal{P}_1$ is the set of all probability measures on the interval $[0,1]$.

    \cite{detstud2002} introduced new ``coordinates'' for the moment space $\mathcal{M}_n^p$ defining  a one to one
 map from the interior of  $\mathcal{M}_n^p$ onto the product space
$(\mathcal{E}_p)^n$, where $\mathcal{E}_p$
 denotes the ``cube''
\begin{align*}
{\cal E}_p =\{ A \in \mathcal{S}_{p}  ~|~ 0_p < A < I_p \}.
\end{align*}
Here and throughout the paper, $0_p$ denotes the $p \times p$ matrix with all elements equal to zero and $I_p$ denotes the $p \times p$ identity matrix.

The new coordinates are called canonical moments  [see \cite{detstud2002}], and they are related to the Verblunsky coefficients,
which have been discussed for matrix measures on the unit circle [see \cite{dampussim2008} and \cite{simon2005a,simon2005b}]. They turn out to be extremely
useful in analyzing the asymptotic properties of the stochastic process defined in \eqref{basicproc}.
The definition of matrix valued
canonical moments
  relies on the introduction of Block-Hankel-matrices:
    \begin{align*}
        \underline{H}_{2k} &=
        \begin{pmatrix}
            M_0 & \cdots & M_k \\
            \vdots & \ddots & \vdots \\
            M_k & \cdots & M_{2k}
        \end{pmatrix}
        &\overline{H}_{2k} =
        \begin{pmatrix}
            M_1 - M_2 & \cdots & M_k - M_{k + 1} \\
            \vdots & \ddots & \vdots \\
            M_k  - M_{k + 1} & \cdots & M_{2k - 1} - M_{2k}
        \end{pmatrix} \\
        \underline{H}_{2k + 1} &=
        \begin{pmatrix}
            M_1 & \cdots & M_{k + 1} \\
            \vdots & \ddots & \vdots \\
            M_{k + 1} & \cdots & M_{2k + 1}
        \end{pmatrix}
        &\overline{H}_{2k + 1} =
        \begin{pmatrix}
            M_0 - M_1 & \cdots & M_k - M_{k + 1} \\
            \vdots & \ddots & \vdots \\
            M_k - M_{k + 1} & \cdots & M_{2k} - M_{2k + 1}
        \end{pmatrix}
    \end{align*}
    which - as in the one dimensional case - can be used to characterize elements of the moment space $\mathcal{M}_n^p$. More precisely,  the vector of matrices
     $M = (M_1, \ldots, M_n)^t \in ( \mathcal{S}_p)^n$  satisfies
         \begin{align*}
        M \in \mathcal{M}_p^n & \text{~ if and only if ~}  \underline{H}_k \ge 0_k, \overline{H}_k \ge 0_k  \text{~ for all ~} k \le n \\
        M \in \inn{\mathcal{M}_p^n} & \text{~ if and only if ~}  \underline{H}_k > 0_k, \overline{H}_k > 0_k  \text{~ for all ~} k \le n~,
    \end{align*}
    where $\inn{\mathcal{M}_p^n}$ denotes the interior of the set $ \mathcal{M}_p^n $.
We now introduce the vectors of matrices
    \begin{align*}
        \underline{h}_{2k}^t &= \big (M_{k + 1}, \ldots, M_{2k}\big ) ~,~
        \underline{h}_{2k - 1}^t = \big (M_k, \ldots, M_{2k - 1}) \\
        \overline{h}_{2k}^t &= \big (M_k - M_{k + 1}, \ldots, M_{2k - 1} - M_{2k}\big ) ~,~
        \overline{h}_{2k - 1}^t = \big (M_k - M_{k + 1}, \ldots, M_{2k - 2} - M_{2k - 1}\big )
    \end{align*}
   and define  the ``extremal'' matrices $    M_1^- = 0_p $,  $M_1^+= I_p $ and
      $  M_2^+ = M_1$ (the phrase  extremal will shortly become clear).     If  $M$ is an element of the interior of the moment space $\mathcal{M}_p^n$,
      the extremal moments of larger order are defined as
    \begin{align*}
        M_n^- &= \underline{h}_{n - 1}^t \underline{H}_{n - 2}^{-1} \underline{h}_{n - 1} \quad \text{ for } n \ge 2 \\
        M_n^+ &= M_{n - 1} - \overline{h}_{n - 1}^t \overline{H}_{n - 2}^{-1} \overline{h}_{n - 1} \quad \text{ for } n \ge 3 .
    \end{align*}
\cite{detstud2002}  showed that the extremal moments provide a convenient tool to characterise the moment space $\mathcal{M}_p^n$. In particular
by considering Schur complements  of $\underline{H}_k$ and  $\overline{H}_k$, they showed that
    \begin{align*}
        (M_1, \ldots, M_n)^t \in \inn{\mathcal{M}_p^n} \text{~ if and only if ~}  M_k^- < M_k < M_k^+  \text{~ for all ~} k \le n ~.
    \end{align*}
 This property is then used to define for  a point
    $ (M_1, \ldots, M_n)^t \in \inn{\mathcal{M}_p^n} $ matrix valued canonical moments   as follows
    \begin{align} \label{defU}
        \mathcal{U}_{i} = (M_i^+ - M_i^-)^{-1/2}(M_i - M_i^-)(M_i^+ - M_i^-)^{-1/2} ~,~i=1, \ldots , n.
    \end{align}
     Given $M_1, \ldots M_{k-1}$, the moment $M_k$ can be calculated from the canonical moment $U_k$ and therefore
 equation \eqref{defU} defines a one to one mapping
\begin{align*}
\varphi_p   : \left \{ \begin{array}{lll}
& \inn{ \mathcal{M}_n^p } \ & \to {\cal E}_p^n  \\
 &(M_{1,n}^p,\dots,M_{n,n}^p)^t & \mapsto \varphi_p (M_{1,n}^p,\dots,M_{n,n}^p) = (U_{1,n}^p,\dots ,U_{n,n}^p)^t\ ,
\end{array} \right .
\end{align*}
from the interior of the moment space onto ${\cal E}_p^n$.

We conclude this section with  a very  interesting and useful relation between canonical moments and determinants  of Hankel  matrices $\underline{H}_{2n} = \det (M_{i+j,2n}^p)_{i,j=0,\ldots ,n}$ corresponding to a
point $ (M_{1,2n}^p ,  \ldots , M_{2n,2n}^p)^t \in  \mbox{Int}( \mathcal{M}_{2n}^p )$, i.e
    \begin{align}\label{produkt_h}
        \det \underline{H}_{2n}
        =& \prod \limits_{i = 1}^n \Big(\det(I_p - \mathcal{U}_{2i - 2,2n}^p) \det(\mathcal{U}_{2i - 1,2n}^p) \det(I_p - \mathcal{U}_{2i - 1,2n}^p)  \det(\mathcal{U}_{2i,2n}^p)\Big)^{n - i + 1} \\
        =& \prod \limits_{i = 1}^n  \big \{ \det(\mathcal{U}_{2i - 1,2n}^p)   \det(I_p - \mathcal{U}_{2i - 1,2n}^p)  \det(\mathcal{U}_{2i,2n}^p)
         \big\} ^{n- i + 1}  (\det(I_p - \mathcal{U}_{2i,2n}^p))^{n - i},\nonumber
    \end{align}
    where $U_0 = 0_p$ [see \cite{detstud2005} for a proof].

\section{The distribution of random Hankel block matrices} \label{section_distribution}
 By identifying a symmetric matrix $M = (m_{i, j})_{i, j = 1}^p \in \mathcal{S}_p$ with the vector containing the entries $m_{i, j}$ for $1 \le i \le j \le p$, we get a  subset of $\mathbb{R}^{p(p + 1)/2}$ with non-empty interior. This identification allows us to integrate on $\mathcal{S}_p$ via the usual Lebesgue-measure
 and to  define distributions on $\mathcal{S}_p$ by specifying their  (Lebesgue-)densities. We are particularly interested in matrix valued Beta distributions
 [see \cite{olkrub1964} or \cite{muir1982}]
  supported on the set $\mathcal{E}_p$ with a density proportional to
      \begin{align} \label{jacbeta}
  f_{  \gamma , \delta }(U) =     (\det U)^{\gamma - e_1} (\det(I_p - U))^{\delta - e_1} I\{0_p < U < I_p\}.
    \end{align}
    where  $e_1 = \frac{p + 1}{2}$ and the parameters $\gamma, \delta$ satisfy $\gamma, \delta > e_1 - 1$.
    These distributions are a special case of the
    Jacobi-beta-ensemble $J\beta E_p(\gamma, \delta)$, which defines a density to be proportional to  \eqref{jacbeta}, where the constant
    $e_\beta$ is given by   $e_\beta = 1 + \frac{\beta}{2} (p - 1)$. The parameter $\beta$ varies when the entries of the matrix are are real ($\beta = 1$), complex ($\beta = 2$) or quaternions ($\beta = 4$) [see for example \cite{arashi2011}]. In the present paper, we will only consider the real case.

  Consider a uniform distribution  on the   $n$-th moment  space $\mathcal{M}_n^p$ defined in \eqref{1.1} and denote by  $\mathcal{U}_{i, n}^p$
  the $i$-th canonical moment in \eqref{defU}, then it is shown in Theorem 3.5 of  \cite{detnag2012a} that
  $\mathcal{U}_{i, n}^p , \ldots , \mathcal{U}_{i, n}^p$ are independent and distributed according to a Jacobi-beta-ensemble, that is
        \begin{align}
            \mathcal{U}_{i, n}^p \sim J\beta E_p(e_1 (n - i + 1), e_1 (n - i + 1)) \label{dist_can_mom}.
        \end{align}
>From this result and formula \eqref{produkt_h} it is obvious that the distribution of determinants of random variables
governed by the  Jacobi-Ensemble will be essential for the following analysis of the process. Our first main result, which is of independent interest,
provides an important tool to determine the distribution of the process $\{ H_n(t)\}_{t\in [0,1]} $ defined in \eqref{basicproc}.
Throughout this paper we will use the notation $M^{[k]} \in \mathbb{R}^{k \times k}$ for   the upper left $k \times k$ subblock of the matrix $M \in \mathbb{R}^{p\times p}$ and
$\beta (\gamma, \delta)$ denotes a Beta distribution on the interval $[0,1]$.

    \begin{thm}[Subblocks of Jacobi-Ensembles] \label{dist_jacobi}
        Assume $p > 1$ and that
         $U \sim J\beta E_p(\gamma, \delta)$. If we denote by $V = U^{[p - 1]}$ the upper left $(p - 1) \times (p - 1)$ subblock of the random matrix  $U$, then
        \begin{align*}
        \Big (V, \frac{\det U}{\det V}, \frac{\det(I_p - U)}{\det(I_{p - 1} - V)} \Big) \stackrel{d}{=} J\beta E_{p - 1}(\gamma, \delta) \otimes (p_{p - 1, 1}, (1-p_{p - 1, 1}) p_{p - 1, 2}),
        \end{align*}
        where the random variables $p_{i, 1} \sim \beta(\gamma - i/2, \delta)$ and $p_{i, 2} \sim \beta(\delta - i/2, i/2)$ are beta-distributed and independent.
    \end{thm}

    \begin{proof}
    We consider the Cholesky decomposition of the matrix
      \begin{align}
            U = \begin{pmatrix}V & B \\ B^t & c\end{pmatrix} =     \begin{pmatrix}
                T^t & 0 \\
                t^t & \sqrt{t_p}
            \end{pmatrix}
                \begin{pmatrix}
                T & t \\
                0 & \sqrt{t_p}
            \end{pmatrix}
            =
            \begin{pmatrix}T^t T & T^t t \\ t^t T & t^t t + t_p\end{pmatrix},\label{dist_jacobi_cholesky}
        \end{align}
        where $V= U^{[p - 1]}$ and   $T$ is an upper triangular   $(p - 1) \times (p - 1)$ matrix with strictly positive entries on the diagonal.
         The matrix $U$ satisfies almost surely the inequality $0_p < U < I_p$.
         As a matrix is positive definite if and only if its main subblock and the correpsonding Schur complement are positve definite, we obtain
for the  terms $V$, $B$ and $c$ in \eqref{dist_jacobi_cholesky}:
        \begin{align}
         & 0_{p - 1} < V \text{ and } c - B^t V^{-1} B > 0 \label{dist_jacobi_schur_1} \\
   & V < I_{p - 1} \text{ and } 1 - c - B^t (I_{p - 1} - V)^{-1} B > 0 \label{dist_jacobi_schur_2}
        \end{align}
 This implies  $c < 1$ and we conclude  $t_p = c - t^t t < 1$. Therefore the random variable
        \begin{align*}
            v = (1 - t_p)^{-1/2}(I_{p - 1} + T(I_{p - 1} - V)^{-1} T^t)^{1/2} t.
        \end{align*}
        is well defined. \\
  We will now determine the joint density of the random variables $V$, $v$ and $t_p$
  (up to a constant).
     For this purpose note that  the equation $c = t^t t + t_p$
        yields the Schur complement of $I_{p - 1} - V$ in $I_p - U$ as
        \begin{align*}
            1 - c - B^t (I_{p - 1} - V)^{-1} B   = &1 - t^t t - t_p - t^t T (I_{p - 1} - V)^{-1} T^t t \\
            =& (1 - t_p)(1 - t^t(I_{p - 1} + T (I_{p - 1} - V)^{-1} T^t)t(1 - t_p)^{-1}) \\
             =& (1 - t_p)(1 - v^tv),
        \end{align*}
   which yields combined with the well-known formula for the    determinant  of a
        Schur-complement
         \begin{align}
         \det (I_p -U) = \det (I_{p-1} -V)    \cdot  (1 - t_p)(1 - v^tv).
       \label{dist_jacobi_schur_3a}
        \end{align}
      Similarly, observing \eqref{dist_jacobi_cholesky}
        \begin{align} \label{dist_jacobi_schur_3b}
            \det(V) (c - B^t V^{-1} B) = \det U = (\det(T) \sqrt{t_p})^2 = \det(T^t T) t_p = \det(V) t_p,
        \end{align}
         which  also  implies
        \begin{align}
            c - B^t V^{-1} B = t_p. \label{dist_jacobi_schur_4}
        \end{align}
        Using (\ref{dist_jacobi_schur_1}), (\ref{dist_jacobi_schur_2}), (\ref{dist_jacobi_schur_3a})  and
         (\ref{dist_jacobi_schur_3b})
      it follows that the density of  $V$, $v$ and $t_p$ is proportional to  the function
        \begin{align}
     g_{  \gamma ,  \delta}(V)         &
            =(\det V)^{\gamma - e_1} t_p^{\gamma - e_1} (\det(I_{p - 1} - V))^{\delta - e_1} ((1 - t_p)(1 - v^tv))^{\delta - e_1}\label{dist_jacobi_density_u}
            \det D^{-1}\\
            &~~~~~~\times  I\{0_{p - 1} < V < I_{p - 1}\} I\{0 < t_p < 1\} I\{v^tv < 1\} \nonumber,
        \end{align}
       where  $\det D^{-1}$ is  the Jacobi-determinant of the corresponding transformation from $U$ to $V$, $v$ and $t_p$.
       As this transformation  leaves the matrix
       $V$  unchanged  we obtain $\det D^{-1}= \det M^{-1}$, where the matrix $M$ is given by
        \begin{align*}
            M = \begin{pmatrix}
            \frac{\partial v}{\partial B} & \frac{\partial v}{\partial c} \\[.3cm]
            \frac{\partial t_p}{\partial B} & \frac{\partial t_p}{\partial c}
        \end{pmatrix}
        \end{align*}
and
        \begin{align*}
            \frac{\partial t_p}{\partial c} &= 1 ~,~
            \frac{\partial t_p}{\partial B} = -2B^t V^{-1}.
        \end{align*}
In order to calculate the remaining elements of the matrix $M$, we simplify the representation of $v$  using the formula
        \begin{align*}
            I_{p - 1} + T(I_{p - 1} - V)^{-1} T^t  = I_{p - 1} + (V^{-1} - I_{p - 1})^{-1}

            = (I_{p - 1} - V)^{-1},
        \end{align*}
        where the second equality stems from an application of the principal axis transform.
   From this equation, (\ref{dist_jacobi_cholesky}) and (\ref{dist_jacobi_schur_4}) we can rewrite the vector $v$ as
        \begin{align*}
            v = (1 - c + B^t V^{-1} B)^{-1/2}(I_{p - 1} - V)^{-1/2} (T^t)^{-1} B.
        \end{align*}
     Standard calculus,   observing this representation and (\ref{dist_jacobi_schur_4})
    now   gives
        \begin{align*}
            \frac{\partial v}{\partial c} &= \frac{1}{2(1 - t_p)} v \\

            \frac{\partial v}{\partial B} &=- \frac{1}{1 -  t_p} v B^t V^{-1} + (1-t_p)^{-1/2} (I_{p - 1} - V)^{-1/2} (T^t)^{-1}~,
        \end{align*}
       and it follows
               \begin{align*}
            \det M = \det\bigg(\frac{\partial v}{\partial B} - \frac{\partial v}{\partial c} \cdot \frac{\partial t_p}{\partial B}\bigg) = (1-t_p)^{-(p - 1)/2} (\det(I_{p - 1} - V))^{-(p - 1)/2} (\det V)^{-1/2}.
        \end{align*}
      From \eqref{dist_jacobi_density_u} we obtain that the joint density of $V$, $v$ and $t_p$
      is proportional to the function
              \begin{align*}
            g_{  \gamma ,  \delta}(V)        & = (\det V)^{\gamma - e_1 + 1/2}( \det(I_{p - 1} - V))^{\delta - e_1 + 1/2} \cdot  t_p^{\gamma - e_1}  (1 - t_p)^{\delta - e_1 + (p - 1)/2} \cdot (1 - v^tv)^{\delta - e_1} \\
            &\qquad \times  I\{0_{p - 1} < V < I_{p - 1}\} I\{0 < t_p < 1\} I\{v^tv < 1\} \\
            &= (\det V)^{\gamma - p/2}( \det(I_{p - 1} - V)^{\delta - p/2} \cdot  t_p^{\gamma - (p - 1)/2 - 1}  (1 - t_p)^{\delta - 1} \cdot (1 - v^tv)^{\delta - (p + 1)/2} \\
            &\qquad  \times  I\{0_{p - 1} < V < I_{p - 1}\} I\{0 < t_p < 1\} I\{v^tv < 1\}
        \end{align*}
        Therefore the random variables $V$, $v$ and $t_p$ are independent with $V \sim J\beta E_{p - 1}(\gamma, \delta)$ and $t_p \sim \beta(\gamma - (p-1)/2, \delta)$.  Obviously,
        \begin{align*}
            \frac{\det U}{\det V} &= t_p ~,~~
            \frac{\det(I_p - U)}{\det(I_{p - 1} - V)}  = (1 - t_p)(1 - v^t v),
        \end{align*}
 and the assertion now follows if we can prove $1 - v^t v \sim \beta(\delta - (p - 1)/2, (p - 1)/2)$, or equivalently $v^t v \sim \beta((p - 1)/2, \delta - (p - 1)/2)$.

        To see this, we will apply Lemma 2.1 in \cite{gupson1997}. Since $v$ has a density proportional to $g(\|v\|_2^2)$ with the function $g(x) = (1 - x)^{\delta - (p + 1)/2}I\{x < 1\}$, the density of $\|v\|_2$ is proportional to
        \begin{align*}
            x^{p - 2} g(x^2) I\{0 < x\} = x^{p - 2}(1 - x^2)^{\delta - (p + 1)/2} I\{0 < x < 1\}
        \end{align*}
        Using a simple substitution, the density of $\|v\|_2^2$ is therefore proportional to
        \begin{align*}
            x^{p/2 - 1}(1 - x)^{\delta - (p + 1)/2} I\{0 < x < 1\} x^{-1/2} = x^{(p - 1)/2 - 1} (1 - x)^{\delta - (p - 1)/2 - 1} I\{0 < x < 1\},
        \end{align*}
        i.e. $v^tv = \|v\|_2^2 \sim \beta((p - 1)/2, \delta - (p - 1)/2)$. This concludes the proof of Theorem~\ref{dist_jacobi}.
    \end{proof}

    \begin{thm} \label{jacobi_subblock}
        Assume $M \sim J\beta E_p(\gamma, \delta)$, and let $M^{[j]}$ denote the $j \times j$ upper left subblock of the matrix $M$.
        There exist independent random variables $p_{0,1}, \ldots , p_{p-1,1}$
        and $p_{1,2}, \ldots , p_{p-1,2}$ with
        \begin{align} \label{distpropa}
            p_{i, 1} &\sim \beta(\gamma - i/2, \delta)  ~,~i=0,\ldots ,p-1, \\
            p_{i, 2} &\sim \beta(\delta - i/2, i/2) ~,~i=1,\ldots ,p-1, \label{distpropb}
        \end{align}
        such that for all real $a, b$ the identity
        \begin{align}
            &\log \Big((\det M^{[j]})^a \cdot (\det(I_j-M^{[j]}))^b\Big) \nonumber \\
            ={}& \log(p_{0, 1}^a (1-p_{0, 1})^b) + \sum \limits_{i = 1}^{j - 1}\left\{\log(p_{i, 1}^a (1-p_{i, 1})^b) + \log(p_{i, 2}^b)\right\} \label{ident_det_jacobi}
        \end{align}
        holds for all $1 \le j \le p$ simultaneously. In particular
        \begin{align*}
  \det M =  \prod_{i = 0}^{p - 1} p_{i, 1}.
        \end{align*}
    \end{thm}
    \begin{proof}
        For $1 \le i \le p - 1$ choose
        \begin{align*}
            p_{0, 1} &= \det M^{[1]} ~,~
            p_{i, 1} = \frac{\det M^{[i + 1]}}{\det M^{[i]}} ~,~
            p_{i, 2} = \frac{\det(I_{i + 1} - M^{[i + 1]})}{(1 - p_{i, 1})\det(I_i - M^{[i]})}~,
        \end{align*}
        then the identity (\ref{ident_det_jacobi}) obviously holds. The statements \eqref{distpropa}  and   \eqref{distpropb}  are  now proved by induction with respect to
         the parameter $p$.    The claim is obviously correct for $p = 1$, since
        \begin{align*}
            p_{0, 1} = \det M^{[1]} = M^{[1]} \sim J\beta E_1(\gamma, \delta) = \beta(\gamma, \delta) ~.
        \end{align*}
        holds.

        Now  assume that \eqref{distpropa}  and   \eqref{distpropb} are satisfied  for $1, \ldots, p - 1$, then an application of Theorem~\ref{dist_jacobi}
        yields
        \begin{align*}
            M^{[p - 1]} &\sim J\beta E_{p - 1}(\gamma, \delta) ,\\
            p_{p - 1, 1} &\sim \beta(\gamma - (p - 1)/2, \delta) ,\\
            p_{p - 1, 2} &\sim \beta(\gamma - (p - 1)/2, (p - 1)/2),
        \end{align*}
      and these three random variables are independent. Since $M^{[p - 1]}\sim J\beta E_{p - 1}(\gamma, \delta)$  and
        \begin{align*}
            (M^{[p]})^{[i]} = (M^{[p - 1]})^{[i]} ~~(  i \le p - 1)
        \end{align*}
                it follows from the induction hypothesis that
        \begin{align*}
            p_{i, 1} \sim \beta(\gamma - i/2, \delta) ~,~  i= 0 \ldots p - 2, \\
            p_{i, 2} \sim \beta(\delta - i/2, i/2)  ~,~  i= 1 \ldots p - 2,
        \end{align*}
        and these random variables are independent. It remains to show that $(p_{i, 1})_{i = 0}^{p - 2}$ and $(p_{i, 2})_{i = 1}^{p - 2}$ are also (jointly) independent of $(p_{p - 1, 1}, p_{p - 1, 2})$. This follows directly from the fact that $(p_{i, 1})_{i = 0}^{p - 2}$ and $(p_{i, 2})_{i = 1}^{p - 2}$ can be written as functions of $M^{[p - 1]}$, which is in turn independent of $(p_{p - 1, 1}, p_{p - 1, 2})$.
    \end{proof}

    \begin{rem} {\rm
    A well-known result  in random matrix theory is the Bartlett decomposition [see  \cite{bart1933}], which states that in the Cholesky decomposition of a Wishart-distributed random matrix the entries are independent and normal resp. \hbox{$\chi^2$-distributed}. A corresponding result for the Jacobi-beta-ensemble was derived by \cite{kshi1961}
           and reads as follows. If $X \sim J\beta E_p(\gamma, \delta)$ has the (random) Cholesky decomposition $X = T^t T$ for some upper triangular matrix $T$, then the diagonal entries $t_{11}, \ldots, t_{pp}$ of $T$ are independent and their squares are beta-distributed, that is  $t_{ii}^2 \sim \beta(\gamma - (i - 1)/2, \delta)$. This result
           is a special case of   Theorem~\ref{jacobi_subblock} that can be obtained for $a = 1$, $b = 0$.

        To see this, denote by $K_i$ the $p \times i$-matrix with $(K_i)_{j, k} = \delta_{jk}$. Then the equation
        \begin{align*}
            X^{[i]} = K_i^t X K_i = K_i^t T^t K_i K_i^t T K_i = (K_i^tT K_i)^t K_i^tT K_i
        \end{align*}
        holds. Noting $\det(K_i^tT K_i) = t_{11} \cdot \ldots \cdot t_{ii}$ we can conclude that
        \begin{align*}
            t_{ii}^2 = \frac{\det(K_i^tT K_i)^2}{\det(K_{i - 1}^t T K_{i - 1})^2} = \frac{\det(X^{[i]})}{\det(X^{[i - 1]})} = p_{i - 1, 1}
        \end{align*}
        are independent random variables with $t_{ii}^2 \sim \beta(\gamma - (i-1)/2, \delta)$.
        }
    \end{rem}

\section{Random Hankel determinant processes}  \label{section_applied_convergence}

    Let    $(M_{1, 2n + 1}^p, \ldots, M_{2n + 1, 2n + 1}^p)$ denote a uniformly distributed random vector on the $(2n + 1)$-th moment space $\mathcal{M}_{2n + 1}^{p_n}$
 and recall the definition of the stochastic process  $\{H_n (t) \}_{t \in [0,1]} $ in \eqref{basicproc}.
   From \eqref{produkt_h}, \eqref{dist_can_mom} and Theorem~\ref{jacobi_subblock} we obtain  the following representation
    \begin{align*}
          H_n( t) ={}& \sum \limits_{i = 1}^{\lfloor nt \rfloor}\Big\{
                \sum \limits_{j = 1}^{p_n  - 1} \left(\log(r_{2n + 1, 2i, j}^{\lfloor nt\rfloor - i}) + \log(r_{2n + 1, 2i - 1, j}^{\lfloor nt\rfloor - i + 1}) \right) \\
                &+\sum \limits_{j = 0}^{p_n- 1}\log(p_{2n + 1, 2i, j}^{\lfloor nt\rfloor - i + 1}(1 - p_{2n + 1, 2i, j})^{\lfloor nt\rfloor - i}) \\
                &+\sum \limits_{j = 0}^{p_n  - 1}\log(p_{2n + 1, 2i - 1, j}^{\lfloor nt\rfloor - i + 1}(1 - p_{2n + 1, 2i - 1, j})^{\lfloor nt\rfloor - i + 1})
            \Big\}~,~
    \end{align*}
    where  the random variables $p_{2n + 1, i, j}$ and $r_{2n + 1, i, j}$ are independent and beta-distributed, that is
    \begin{align}
        p_{2n + 1, i, j} &\sim \beta\Big (\frac{p_n + 1}{2}(2n - i + 2) - j/2, \frac{p_n + 1}{2}(2n - i + 2)\Big ) \label{beta_param_1} \\
        r_{2n + 1, i, j} &\sim \beta\Big (\frac{p_n + 1}{2}(2n - i + 2) - j/2, j/2\Big  ). \label{beta_param_2}
    \end{align}
    In the following discussion, we will consider a more general process. When viewing the Hankel-determinant as a function of the canonical moment matrices $\mathcal{U}_{i, 2n}^{p_n}$ [see equation~\eqref{produkt_h}], we can not only vary the dimension $n$ of the Hankel-matrix, but also the size $p_n$ of the canonical moment matrices in \eqref{produkt_h}. To this extent we introduce a new parameter $s \in (0, 1]$ and consider the upper left $\lfloor p_n s \rfloor \times \lfloor p_n s \rfloor$ subblocks of the canonical moment matrices. According to Theorem~\ref{jacobi_subblock}, the distribution of the logarithm of the corresponding Hankel-determinant can be written as
       \begin{align}
        \label{def_h}
        \begin{split}
          H_n(s, t) ={}&\sum \limits_{i = 1}^{\lfloor nt \rfloor}\Big\{
                \sum \limits_{j = 1}^{\lfloor p_n s \rfloor - 1} \left(\log(r_{2n + 1, 2i, j}^{\lfloor nt\rfloor - i}) + \log(r_{2n + 1, 2i - 1, j}^{\lfloor nt\rfloor - i + 1}) \right) \\
                &+\sum \limits_{j = 0}^{\lfloor p_ns \rfloor - 1}\log(p_{2n + 1, 2i, j}^{\lfloor nt\rfloor - i + 1}(1 - p_{2n + 1, 2i, j})^{\lfloor nt\rfloor - i}) \\
                &+\sum \limits_{j = 0}^{\lfloor p_n s \rfloor - 1}\log(p_{2n + 1, 2i - 1, j}^{\lfloor nt\rfloor - i + 1}(1 - p_{2n + 1, 2i - 1, j})^{\lfloor nt\rfloor - i + 1})
            \Big\}~,~
        \end{split}
    \end{align}
where we use the convention  $H_n(0, t) = 0$ and the process in  \eqref{basicproc} is obtained as $H_n( t) = H_n(1, t)  $. In the following discussion we will investigate the weak convergence of the process $\{H_n(s, t)\}_{s, t \in [0, 1]}$ as $n, p_n \to \infty$.

For this
purpose we state a general result with sufficient conditions for  the weak convergence of  a process of the form \eqref{def_h}, which might be of independent interest.       The proof can be found in the Appendix~\ref{sec6}.
        \begin{thm} \label{limit_thm}
       For each $n\in \mathbb{N} $   let  $T_n$  be a finite set, let $\{ X_n(i)~|~i \in T_n \}$ be real valued random variables  and let  $g_n: T_n \times [0, 1]^k
       \to \mathbb{R}$ be a  real-valued function. Consider a process of the form
    \begin{align*}
        Z_n(t_1, \ldots, t_k) = \sum \limits_{i \in T_n} g_n(i, t_1, \ldots, t_k) X_n(i)
            \end{align*}
 and suppose that the following assumptions are satisfied
    \begin{enumerate}[label=\textbf{(C\arabic*)}]
        \item The random variables $(X_n(i))_{i \in T_n}$ are independent. \label{cond_indep}
        \item $g_n$ is right-continuous in each of the last $k$ components. \label{cond_cont}
        \item $\e{X_{n}(i)} = 0$. \label{cond_center}
        \item $\e{X_{n}^{2k + 2}(i)} \le C \e{X_{n}^2(i)}^{k + 1} < \infty$ for some universal constant $C > 1$. \label{cond_moments}
        \item $\sup \limits_{i \in T_n} g_n^2(i, t) \var{X_n(i)} \xrightarrow{n \to \infty} 0$ for all fixed $t \in [0, 1]^k$. \label{cond_var_bound}
        \item There exists a function $f:[0, 1]^{2k} \to \mathbb{R}$ such that for all $s, t \in [0, 1]^k$ \label{cond_var_conv}
            \begin{align*}
                \cov{Z_n(s)}{Z_n(t)} = \sum \limits_{i \in T_n} g_n(i, s)g_n(i, t)\var{X_n(i)} \xrightarrow{n \to \infty} f(s, t).
            \end{align*}
        \item There are sequences $h_n^{(j)} \xrightarrow{n \to \infty} 0$ for $1 \le j \le k$ such that for any two vectors $T = (t_1, \ldots, t_k)$ and $T' = (t_1, \ldots, t_j', \ldots, t_k)$ the inequality
            \begin{align*}
                \var{Z_n(T) - Z_n(T')} \le C(|t_j - t_j'| + h_n^{(j)})
            \end{align*}
            holds.\label{cond_cov}
        \item For any $t_j \le t_j' \le t_j'' \le t_j + h_n^{(j)}$ at least one of the equations
            \begin{align*}
                g_n(i, t_1, \ldots, t_j, \ldots, t_k) &= g_n(i, t_1, \ldots, t_j', \ldots, t_k)\\
                &\text{or}\\
                g_n(i, t_1, \ldots, t_j', \ldots, t_k) &= g_n(i, t_1, \ldots, t_j'', \ldots, t_k)
            \end{align*}
            holds. \label{cond_lattice}
    \end{enumerate}
  Then the process $Z_n$ converges weakly in $l^\infty([0, 1]^k)$ to a centered, continuous Gaussian process with covariance kernel $f$.
    \end{thm}

    \begin{thm}\label{convergence_hn}
        If $p_n \xrightarrow{n \to \infty} \infty$, then
        \begin{align*}
            \{ \widetilde{H}_n(s, t) \}_{s,t \in [0,1]} := \frac{1}{\sqrt{n}} \big \{ H_n (s, t) - \mathbb{E}[H_n(s, t) ]\big  \}_{s,t \in [0,1]} \implies \{ \mathcal{G}(s, t) \}_{s,t \in [0,1]}
        \end{align*}
      in $l^\infty([0, 1]^2)$,  where  $\{ H_n(s, t) \}_{s,t \in [0,1]} $ is the process defined in (\ref{def_h})
                and $\mathcal{G}$ is a centered continuous Gaussian process with covariance kernel
        \begin{align*}
            \cov{\mathcal{G}(s_1, t_1)}{\mathcal{G}(s_2, t_2)} = \frac{(s_1 \wedge s_2)^2 c(t_1, t_2)}{2}.
        \end{align*}
        The function $c$ is given by
        \begin{align*}
            c(t_1, t_2) = \int_0^{t_1 \wedge t_2} \frac{(t_1 - x)(t_2 - x)}{(1 - x)^2} dx.
        \end{align*}
    \end{thm}
    \begin{proof}
        We use the decomposition
                \begin{align*}
            \widetilde{H}_n = A_n + B_n + C_n + D_n + E_n
        \end{align*}
 of  the process $\widetilde{H}_n$,       where the processes on the right-hand side are defined by
        \begin{align*}
            A_n ={}& \frac{1}{\sqrt{n}}\sum \limits_{i = 1}^{\lfloor nt \rfloor - 1}\sum \limits_{j = 1}^{\lfloor p_ns \rfloor - 1}(\lfloor nt \rfloor - i)\left(\log(r_{2n + 1, 2i, j}) - \e{\log(r_{2n + 1, 2i, j})}\right) \\
                %
            B_n ={}& \frac{1}{\sqrt{n}}\sum \limits_{i = 1}^{\lfloor nt \rfloor}\sum \limits_{j = 1}^{\lfloor p_ns \rfloor - 1}(\lfloor nt \rfloor - i + 1)\left(\log(r_{2n + 1, 2i - 1, j}) - \e{\log(r_{2n + 1, 2i - 1, j})}\right) \\
                %
            C_n ={}& \frac{1}{\sqrt{n}}\sum \limits_{i = 1}^{\lfloor nt \rfloor - 1}\sum \limits_{j = 0}^{\lfloor p_ns \rfloor - 1} (\lfloor nt \rfloor - i)\big\{\log(p_{2n + 1, 2i, j}(1 - p_{2n + 1, 2i, j}))\\
            &\qquad- \e{\log(p_{2n + 1, 2i, j}(1 - p_{2n + 1, 2i, j}))}\big\} \\
                %
            D_n ={}& \frac{1}{\sqrt{n}}\sum \limits_{i = 1}^{\lfloor nt \rfloor}\sum \limits_{j = 0}^{\lfloor p_ns \rfloor - 1} (\lfloor nt \rfloor - i + 1)\big\{\log(p_{2n + 1, 2i - 1, j}(1 - p_{2n + 1, 2i - 1, j})) \\
            &\qquad- \e{\log(p_{2n + 1, 2i - 1, j}(1 - p_{2n + 1, 2i - 1, j}))}\big\} \\
                %
            E_n ={}&\frac{1}{\sqrt{n}} \sum \limits_{i = 1}^{\lfloor nt \rfloor}\sum \limits_{j = 0}^{\lfloor p_ns \rfloor - 1} \left(\log(p_{2n + 1, 2i, j})
            - \e{\log(p_{2n + 1, 2i, j})}\right)
        \end{align*}
   and the random variables $p_{2n+1,i,j} , r_{2n+1,i,j}$ are independent and beta-distributed, as specified in \eqref{beta_param_1} and \eqref{beta_param_2} respectively. We will now apply Theorem \ref{limit_thm} to each of these processes to prove
        \begin{align}
            A_n &\implies \mathcal{G}'   \label{an}\\
            B_n &\implies \mathcal{G}'  \label{bn} \\
            C_n &\implies 0  \label{cn}  \\
            D_n &\implies 0   \label{dn}\\
            E_n &\implies 0,  \label{en}
        \end{align}
        where $\mathcal{G}'$ is a continuous centered Gaussian process with covariance kernel
        \begin{align*}
            \cov{\mathcal{G}'(s_1, t_1)}{\mathcal{G}'(s_2, t_2)} = \frac{(s_1 \wedge s_2)^2 c(t_1, t_2)}{4}.
        \end{align*}
        Since $A_n$ and $B_n$ are independent  it follows that $                   (A_n, B_n) \implies (\mathcal{G}', \mathcal{G}'') $, where $\mathcal{G}''$ is an
        independent copy of $\mathcal{G}'$ (c.f. Example 1.4.6 in \cite{vanwell1996}),  and the continuous mapping theorem
        implies $A_n + B_n \implies \mathcal{G}' + \mathcal{G}'' \stackrel{d}{=} \mathcal{G}$. The assertion of Theorem \ref{convergence_hn}  now follows from
         Slutsky's lemma.
        We will omit the proof of \eqref{bn}  and \eqref{dn} because  the arguments are similar as in the proof of
    \eqref{an}  and \eqref{cn}, respectively.  \bigskip

     {   \bf Proof of \eqref{an}:}     We can represent the process  $A_n$ in a  form such that  Theorem \ref{limit_thm} is aplicable, that is
        \begin{align} \label{hol1}
            A_n(s, t) = \sum \limits_{(i,j)  \in T_n} g_n((i,j) , s, t) X_n((i,j) )~,
        \end{align}
        where
        \begin{align}
            T_n &= \{1, \ldots, n - 1\} \times \{1, \ldots, p_n - 1\}  \nonumber \\
            g_n((i, j), s, t) &= \frac{1}{\sqrt{n}} (\lfloor nt \rfloor - i) I\{i \le \lfloor nt \rfloor - 1\} I\{j \le \lfloor p_n s\rfloor - 1\}
            \nonumber \\
            X_n((i, j)) &= \log(r_{2n + 1, 2i, j}) - \e{\log(r_{2n + 1, 2i, j})} \label{hol2}
        \end{align}
        It is obvious that $A_n$ satisfies the conditions \ref{cond_indep}, \ref{cond_cont} and \ref{cond_center}. Condition \ref{cond_moments} is proved in Theorem \ref{mom_high} in Appendix~\ref{betamom} (note that the parameters of the distribution of $r_{2n + 1, i, j}$ are bounded from below by $\tfrac{1}{2}$).

        By \eqref{form_kappa1} from Appendix~\ref{betamom} the variance of the logarithm of a beta distributed  random variable $X \sim \beta(a, b)$ can be calculated as
        \begin{align*}
            \var{\log X} = \psi_1(a) - \psi_1(a + b),
        \end{align*}
        where
        \begin{equation} \label{polgma}
        \psi_k(x) = \frac{d^{k + 1}}{dx^{k + 1}} \log \Gamma(x)
        \end{equation}
denotes the Polygamma function.
     An application of formula (\ref{bound_psi_diff}) from Appendix \ref{appendix_polygamma}  shows  that
        \begin{align*}
            &g_n^2((i, j), s, t) \var{X_n(i, j)} \\
            \le{}&  \frac{3 (n - i)^2j/2}{n \left(\frac{p_n + 1}{2}(2n - 2i + 2) - j/2 \right)\frac{p_n + 1}{2}(2n - 2i + 2)} \\
            \le{}&   \frac{3(n - i)^2j/2}{n \left((p_n + 1)(n - i + 1/2)  + (p_n + 1 - j)/2 \right)( p_n + 1) (n - i + 1)} \\
            \le{}& \frac{3j/2}{n \left(p_n + 1\right)^2} \le  \frac{3}{n \left(p_n + 1\right)} \xrightarrow{n \to \infty} 0~,
        \end{align*}
   and therefore  condition \ref{cond_var_bound} is also satisfied.
  To see that condition \ref{cond_var_conv} holds,    define for  $s_1, s_2, t_1, t_2 \in [0, 1]$  the minima  $s = s_1 \wedge s_2$ and $t = t_1 \wedge t_2$. By (\ref{var_first_order}) we have a decomposition
        \begin{align*}
            \cov{A_n(s_1, t_1)}{A_n(s_2, t_2)} = S_n + R_n,
        \end{align*}
        where
        \begin{align*}
            S_n = \sum \limits_{i = 1}^{\lfloor nt \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s\rfloor - 1} \frac{1}{n}\frac{(\lfloor nt_1\rfloor - i) (\lfloor nt_2\rfloor - i) j/2}{\left(\frac{p_n + 1}{2} (2n - 2i + 2) - j/2\right) \frac{p_n + 1}{2} (2n - 2i + 2)},
        \end{align*}
        and the remainder $R_n$ satisfies the inequality
        \begin{align*}
            |R_n| \le \sum \limits_{i = 1}^{\lfloor nt \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s \rfloor - 1} \frac{4}{n} \frac{(\lfloor nt_2 \rfloor - i)(\lfloor nt_2 \rfloor - i)}{\left(\frac{p_n + 1}{2}(2n - 2i + 2) - j/2\right)^2}.
        \end{align*}
        Now note that
        \begin{align*}
            S_n ={}& \sum \limits_{i = 1}^{\lfloor nt \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s\rfloor - 1} \frac{1}{n}\frac{(\lfloor nt_1\rfloor - i) (\lfloor nt_2\rfloor - i) j/2}{\left(\frac{p_n + 1}{2} (2n - 2i + 2) - j/2\right) \frac{p_n + 1}{2} (2n - 2i + 2)} \\
                %
            \ge{}&\sum \limits_{i = 1}^{\lfloor nt \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s\rfloor - 1} \frac{1}{n}\frac{(nt_1 - i - 1) (nt_2 - i - 1)j/2}{\left(\frac{p_n + 1}{2} (2n - 2i + 2)\right)^2} \\
                %
            ={}&\frac{1}{n}
            \sum \limits_{j = 1}^{\lfloor p_n s \rfloor - 1} \frac{j}{2(p_n + 1)^2}   \sum \limits_{i = 1}^{\lfloor nt \rfloor - 1}\frac{(nt_1 - i - 1) (nt_2 - i - 1)}{(n - i + 1)^2} \\
                %
            ={}& \frac{1}{4} (s^2 + O(p_n^{-1}))\left(c(t_1, t_2) + O(n^{-1})\right) \xrightarrow{n \to \infty} \frac{(s_1 \wedge s_2)^2 c(t_1, t_2)}{4} .
        \end{align*}
Moreover,
        \begin{align*}
            S_n ={}&\sum \limits_{i = 1}^{\lfloor nt\rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s \rfloor - 1}\frac{1}{n} \frac{(\lfloor nt_1\rfloor - i) (\lfloor nt_2\rfloor - i) j/2}{\left(\frac{p_n + 1}{2} (2n - 2i + 2) - j/2\right) \frac{p_n + 1}{2} (2n - 2i + 2)} \\
            \le{}&\sum \limits_{i = 1}^{\lfloor nt \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s \rfloor - 1} \frac{1}{n}\frac{(nt_1 - i) (nt_2 - i)j/2}{\left(\frac{p_n + 1}{2} (2n - 2i)\right)^2} \\
            ={}& \Big (\frac{s^2}{4} + O(p_n^{-1})\Big )\left(c(t_1, t_2) + O(n^{-1})\right) \xrightarrow{n \to \infty} \frac{(s_1 \wedge s_2)^2 c(t_1, t_2)}{4}~,
        \end{align*}
       which gives
        \begin{align*}
            S_n \xrightarrow{n \to \infty} \frac{(s_1 \wedge s_2)^2 c(t_1, t_2)}{4}.
        \end{align*}
Finally, the remainder $R_n$ vanishes asymptotically, that is
        \begin{align*}
            |R_n| \le{}&\sum \limits_{i = 1}^{\lfloor nt \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s \rfloor - 1} \frac{4}{n} \frac{(\lfloor nt_2 \rfloor - i)(\lfloor nt_2 \rfloor - i)}{\left(\frac{p_n + 1}{2}(2n - 2i + 2) - j/2\right)^2} \\
            \le{}& \sum \limits_{i = 1}^{\lfloor nt \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s \rfloor - 1}\frac{4}{n} \frac{(n - i)^2}{((p_n + 1)(n - i + 1/2))^2}
            \le{} \frac{4}{p_n} \xrightarrow{n \to \infty} 0,
        \end{align*}
        which proves condition \ref{cond_var_conv}.
        To show   \ref{cond_cov} note that
        \begin{align}
             \var{\log r_{2n, 2i, j}}
            \le{}  5 \frac{j/2}{\left(\frac{p_n + 1}{2}(2n - 2i + 2) - j/2\right)^2}
            \le{}& \frac{5j}{((p_n + 1)(n - i))^2} \label{u_bnd_1}
        \end{align}
        holds by the upper bound (\ref{bound_psi_diff}) in Appendix \ref{appendix_polygamma}.     Assume now that $0 \le t_1 < t_2 \le 1$ and $0 \le s \le 1$ are real numbers. Then for $X = (s, t_1)$ and $Y = (s, t_2)$
        we have
        \begin{align*}
            \var{Z_n(Y) - Z_n(X)}

                %
            \le{}& \sum \limits_{i = 1}^{\lfloor nt_1 \rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s\rfloor - 1} \frac{5}{n}(\lfloor nt_2 \rfloor - \lfloor nt_1 \rfloor)^2\frac{j}{(p_n + 1)^2 (n - i)^2} \\
            &+ \sum \limits_{i = \lfloor nt_1 \rfloor}^{\lfloor n t_2\rfloor - 1} \sum \limits_{j = 1}^{\lfloor p_n s\rfloor - 1} \frac{5}{n}(\lfloor nt_2 \rfloor - i)^2 \frac{j}{(p_n + 1)^2 (n - i)^2} \\
                %
            \le{}& \sum \limits_{i = 1}^{\lfloor nt_1 \rfloor - 1} \frac{5}{n}(\lfloor nt_2 \rfloor - \lfloor nt_1 \rfloor)^2 \frac{1}{(n - i)^2}
            + \sum \limits_{i = \lfloor nt_1 \rfloor}^{\lfloor n t_2\rfloor - 1} \frac{5}{n}(\lfloor nt_2 \rfloor - i)^2 \frac{1}{(n - i)^2} \\
                %
            \le{}& \frac{(\lfloor nt_2 \rfloor - \lfloor nt_1 \rfloor)^2}{n^2}\sum \limits_{i = 1}^{\lfloor n t_1 \rfloor - 1} \frac{5}{n}\frac{1}{(1 - i/n)^2} + \sum \limits_{i = \lfloor nt_1 \rfloor}^{\lfloor n t_2\rfloor - 1} \frac{5}{n} \\
                %

                %

                %
            \le{}& 10 \Big (t_2 - t_1 + \frac{1}{n}\Big ).
        \end{align*}
        For the increments in the second coordinate, let $0 \le s_1 < s_2 \le 1$ and $0 \le t \le 1$ be real numbers and set $X = (s_1, t)$, $Y = (s_2, t)$, then
        \begin{align*}
            \var{Z_n(Y) - Z_n(X)}
            & =\sum \limits_{i = 1}^{\lfloor n t\rfloor - 1} \sum \limits_{j = \lfloor p_n s_1 \rfloor}^{\lfloor p_n s_2\rfloor - 1} \frac{1}{n}(\lfloor nt \rfloor - i)^2\var{\log r_{2n, 2i, j}}\\
                %
         &    \le{}\sum \limits_{i = 1}^{\lfloor n t\rfloor - i} \sum \limits_{j = \lfloor p_n s_1 \rfloor}^{\lfloor p_n s_2\rfloor - 1} \frac{5}{n}\frac{j}{(p_n + 1)^2}
            \le \frac{5}{p_n + 1}\sum\limits_{j = \lfloor p_n s_1 \rfloor}^{\lfloor p_n s_2\rfloor - 1} 1\\
                %
          &  \le{} 5\Big (s_2 - s_1 + \frac{1}{p_n}\Big ).
        \end{align*}
        Therefore condition  \ref{cond_cov} is also satisfied with $h_n^{(1)} = \frac{1}{p_n}$ and $h_n^{(2)} = \frac{1}{n}$. It is obvious that \ref{cond_lattice} holds for these sequences and the assertion \eqref{an}  follows from Theorem \ref{limit_thm}. \bigskip

    {\bf Proof of    \eqref{cn}:}
    The process     $C_n$ can be decomposed in a similar way as $A_n$ in \eqref{hol1}, where the random variables
    $X_n((i,j) )$  in \eqref{hol2} are now defined by
   \begin{align*}
        X_n((i,j) ) = \log\big(p_{2n + 1, 2i, j}(1 - p_{2n + 1, 2i, j})\big) - \e{\log\big(p_{2n + 1, 2i, j}(1 - p_{2n + 1, 2i, j})\big)}.
   \end{align*}

   It is again obvious that  \ref{cond_indep}, \ref{cond_cont} and \ref{cond_center} hold. Condition \ref{cond_moments}
is a consequence of Theorem \ref{mom_high_2} in Appendix \ref{betamom}.

        For a  proof of   \ref{cond_var_bound} note that  by  formula (\ref{form_kappa2})  in Appendix \ref{betamom}
        the variance of $\log(X(1-X))$ for a    random variable $X \sim \beta(a, b)$  can be calculated as
        \begin{align*}
            \var{\log(X(1-X))} = \psi_1(a) + \psi_1(b) - 4 \psi_1(a + b)~.
        \end{align*}
An application of (\ref{bnd_var}) now  gives
        \begin{align}
            \var{\log(p_{2n + 1, 2i, j}(1 - p_{2n + 1, 2i, j}))}
            \le{}& \Big(6 + \frac{(j/2)^2}{\left(\frac{p_n + 1}{2}(2n - 2i + 2) - \frac{j}{2}\right)}\Big) \frac{1}{\left(\frac{p_n + 1}{2}(2n - 2i + 2)\right)^2} \nonumber\\
            \le{}& \Big (6 + \frac{j^2}{(p_n + 1)(n - i)}\Big ) \frac{1}{((p_n + 1)(n - i))^2} \nonumber\\
            ={}& \frac{6}{((p_n + 1)(n - i))^2} + \frac{j^2}{((p_n + 1)(n - i))^3} \nonumber\\
            \le{}& \frac{6(j + 1)}{((p_n + 1)(n - i))^2}, \label{u_bnd_2}
        \end{align}
    and condition        \ref{cond_var_bound} follows from the inequality
        \begin{align*}
            \frac{1}{n} (\lfloor nt\rfloor - i)^2\var{\log(p_{2n + 1, 2i, j}(1 - p_{2n + 1, 2i, j})}    \le{}
            \frac{1}{n} (\lfloor nt\rfloor - i)^2 \frac{6(j + 1)}{((p_n + 1)(n - i))^2} \le{}\frac{6}{np_n} .

        \end{align*}
    For a proof of assumption     \ref{cond_var_conv} note that
        \begin{align*}
            \var{C_n(s, t)}

                %
            \le{}& \sum \limits_{i = 1}^{n - 1} \sum \limits_{j = 0}^{p_n - 1}\frac{1}{n} \Big (6 + \frac{j^2}{((p_n + 1)(n - i))^2}\Big ) \frac{1}{(p_n + 1)^2} \\
                %
            \le{}& \frac{6}{p_n + 1} + \sum \limits_{i = 1}^{n - 1} \sum \limits_{j = 0}^{p_n - 1}\frac{1}{n} \frac{1}{(p_n + 1)i} \frac{j^2}{(p_n+ 1)^2} \\
                %
            \le{}& \frac{6}{p_n + 1} + \frac{\log(n) + 1}{n} \xrightarrow{n \to \infty} 0.
        \end{align*}
        By the Cauchy-Schwarz-inequality this implies
        \begin{align*}
            \cov{C_n(s_1, t_1)}{C_n(s_2, t_2)} \xrightarrow{n \to \infty} 0.
        \end{align*}
   Assumption          \ref{cond_cov} follows by similar calculations as given for the proof of \eqref{an} (note the similarity between (\ref{u_bnd_1}) and (\ref{u_bnd_2})). More specifically, the inequalities
        \begin{align*}
            \var{C_n(s, t_1) - C_n(s, t_2)} &\le 24 \Big (|t_2 - t_1| + \frac{1}{n}\Big ) \\
            \var{C_n(s_1, t) - C_n(s_2, t)} &\le 12 \Big( |s_2 - s_1| + \frac{1}{p_n}\Big )
        \end{align*}
        hold for any real numbers $s, t \in [0, 1]$, $0 \le s_1 < s_2 \le 1$ and $0 \le t_1 \le t_2 \le 1$. Finally
        condition    \ref{cond_lattice} is obvious with $h_n^{(1)} = \frac{1}{p_n}$, $h_n^{(2)} = \frac{1}{n}$  and the assertion \eqref{cn} follows
        from Theorem  \ref{limit_thm}. \bigskip

      {\bf Proof of \eqref{en}:}   We use again Theorem  \ref{limit_thm} to prove the assertion.
       Conditions \ref{cond_indep} -- \ref{cond_moments} hold by similar arguments as in the proof of $A_n$.
     Observing the inequality
        \begin{align}
            \var{\log(p_{2n + 1, 2i, j}) }
            \le{}& 5 \frac{\frac{p_n + 1}{2}(2n - 2i + 2)}{\left(\frac{p_n + 1}{2}(2n - 2i + 2) - j/2\right)\left((p_n + 1)(2n - 2i + 2) - j/2\right)}\nonumber\\
            \le{}& 5 \frac{(p_n + 1)(n - i + 1)}{\left((p_n + 1)(n - i + 1/2)\right)\left((p_n + 1)(2n - 2i + 1)\right)} \nonumber\\
            \le{}& 5 \frac{n - i + 1}{(p_n + 1)(n - i + 1/2)^2} \le \frac{5}{(p_n + 1)(n - i + 1/2)} \label{bnd_rem}
        \end{align}
   (which follows from      (\ref{bound_psi_diff})) condition   \ref{cond_var_bound} is obviously satisfied.
        To prove \ref{cond_var_conv}, note that it follows from (\ref{bnd_rem})
        \begin{align*}
            \var{E_n(s, t)} \le \frac{1}{n}\sum \limits_{i = 1}^{n} \sum \limits_{j = 0}^{p_n - 1} \frac{5}{(p_n + 1)(n - i + 1/2)} = O\left(\frac{\log(n)}{n}\right) \xrightarrow{n \to \infty} 0.
        \end{align*}
        Again, by the Cauchy-Schwarz-inequality, this implies
        \begin{align*}
            \cov{E_n(s_1, t_1)}{E_n(s_2, t_2)} \xrightarrow{n \to \infty} 0.
        \end{align*}
For a proof of      \ref{cond_cov} let $0 \le s_1 < s_2 \le 1$ and $0 \le t \le 1$, then we obtain for
$X = (s_1, t)$ and $Y = (s_2, t)$ from   (\ref{bnd_rem}) the estimate
        \begin{align*}
            \var{E_n(X) - E_n(Y)} \le \frac{1}{n} \sum \limits_{i = 1}^{\lfloor nt \rfloor}\sum \limits_{j = \lfloor p_n s_1\rfloor}^{\lfloor p_n s_2 \rfloor - 1}\frac{5}{(p_n + 1)(n - i + 1/2)} \le 10 \Big(s_2 - s_1 + \frac{1}{p_n}\big )~,
        \end{align*}
        and similarly for  $0 \le t_1 < t_2 \le 1$, $0 \le s \le 1$,  $X = (s, t_1)$ and $Y = (s, t_2)$
        \begin{align*}
            \var{E_n(X) - E_n(Y)} \le \frac{1}{n} \sum \limits_{i = \lfloor nt_1\rfloor + 1}^{\lfloor nt_2 \rfloor}\sum \limits_{j = 0}^{\lfloor p_n s \rfloor - 1}\frac{5}{(p_n + 1)(n - i + 1/2)} \le 10 \Big (t_2 - t_1 + \frac{1}{n}\Big ).
        \end{align*}
      Finally  assumption    \ref{cond_lattice} is obvious with $h_n^{(1)} = \frac{1}{p_n}$ and $h_n^{(2)} = \frac{1}{n}$.
    \end{proof}

    \begin{cor}
        If $p_n \xrightarrow{n \to \infty} \infty$, then
        \begin{align*}
            \{ \widetilde{H}_n( t) \}_{t \in [0,1]} := \frac{1}{\sqrt{n}} \big \{ H_n ( t) - \mathbb{E}[H_n(t) ]\big  \}_{t \in [0,1]} \implies \{ \mathcal{G} (t) \}_{t \in [0,1]}  ,
        \end{align*}
       in $l^\infty([0, 1])$   where $\{ H_n( t) \}_{t \in [0,1]} $ is the process defined in (\ref{basicproc})
        and $\{ \mathcal{G}( t) \}_{t \in [0,1]} $ is a centered continuous Gaussian process with covariance kernel
        \begin{align*}
            \cov{\mathcal{G}( t_1)}{\mathcal{G}( t_2)} = \frac{1}{2}\int_0^{t_1 \wedge t_2} \frac{(t_1 - x)(t_2 - x)}{(1 - x)^2} dx.
        \end{align*}
    \end{cor}

    \medskip
We conclude with a Glivenko-Cantelli type Theorem.  \medskip

    \begin{thm} \label{conv_as}
        If $p_n \xrightarrow{n \to \infty} \infty$, we have for any $\varepsilon > 0$
        \begin{align*}
            n^{-\varepsilon} \|\widetilde H_n\| \xrightarrow{a.s.} 0.
        \end{align*}
    \end{thm}
    \begin{proof}
        We will prove a more general result, namely that for a process $Z_n$ satisfying assumptions \ref{cond_indep} -- \ref{cond_lattice} we have $a_n \|Z_n\|_{\infty} \xrightarrow{a.s.} 0$ for all sequences $a_n$ that satisfy $a_n = O(n^{-(1 + \delta)/(2k + 2)})$ for some $\delta > 0$. In order to prove this, note that by (\ref{ineq_dev}) and (\ref{ineq_fn}) we can apply Theorem~\ref{sup_ineq} from Appendix~\ref{sec6} to conclude that
        \begin{align*}
            \p{\sup \limits_{s, k \in [0, 1]^k} |Z_n(s) - Z_n(t)| > \lambda} \le C'' \lambda^{-(2k+2)}
        \end{align*}
        holds for all $\lambda > 0$, where $C''$ is some constant independent of $\lambda$. Combined with \ref{cond_moments} and \ref{cond_var_conv} this yields
        \begin{align*}
            &\sum \limits_{n = 1}^\infty \p{a_n \|Z_n\|_\infty > \lambda}\\
            \le{}& \sum \limits_{n = 1}^\infty \p{\sup \limits_{s, t \in [0, 1]^k} |Z_n(s) - Z_n(t)| > a_n^{-1} \lambda /2} + \p{|Z_n(0)| > a_n^{-1}\lambda /2} \\
            \le{}& \sum \limits_{n = 1}^\infty \left(C'' + C(f(0, 0) + o(1))^{k + 1}\right)a_n^{2k + 2}(\lambda/2)^{-(2k + 2)} \\
            ={}& \left(C'' + C(f(0, 0) + o(1))^{k + 1}\right)(\lambda/2)^{-(2k + 2)} \sum \limits_{n = 1}^\infty O(n^{-(1+\delta)}) < \infty.
        \end{align*}
        By the Borel-Cantelli Lemma, this implies $a_n \|Z_n\|_\infty \xrightarrow{a.s.} 0$.

        To prove the statement of Theorem~\ref{conv_as}, set $k = \lceil\varepsilon^{-1} + 1 \rceil$, $\delta = 1$ and $a_n = n^{-1/(k+1)}$. We can now define a $k$-dimensional partial sum process $\widehat{H}_n$ via
        \begin{align*}
            \widehat{H}_n(s_1, \ldots, s_k) = \widetilde{H}_n(s_1, s_2).
        \end{align*}
        Similar to the proof of Theorem~\ref{convergence_hn}, we can decompose $\widehat{H}_n$ into a sum of processes that satisfy the conditions \ref{cond_indep} -- \ref{cond_lattice} and from the previous result we can conclude
        \begin{align*}
            n^{-\varepsilon} \|\widetilde{H}_n\|_\infty \le a_n \|\widehat{H}_n\|_\infty \xrightarrow{a.s.} 0.
        \end{align*}
    \end{proof}

    \begin{cor}[Law of large numbers] \label{thm_lln}
        If $p_n \xrightarrow{n \to \infty} \infty$, we have
        \begin{align*}
            \frac{H_n(s, t)}{n p_n} \xrightarrow[\rm uniformly]{a.s.} -\frac{s^2}{2}(t + (1 - t) \log(1 - t))
        \end{align*}
        for $n \to \infty$.
    \end{cor}
    \begin{proof}
        By Theorem~\ref{conv_as} we know
        \begin{align*}
            \frac{\| H_n - \e{H_n}\|_\infty}{n p_n} = p_n^{-1} n^{-1/2} \|\tilde{H}_n\|_\infty \xrightarrow{a.s.} 0
        \end{align*}
        The assertion now follows from Lemma~\ref{lemma_cumulant1} and Lemma~\ref{lemma_cumulant2} of the following section, which yield
        \begin{align*}
            \frac{\e{H_n}}{n p_n} \xrightarrow[\text{uniformly}]{n \to \infty} -\frac{s^2}{2} \int_0^t \frac{t - x}{1 - x} \, dx = -\frac{s^2}{2}(t + (1 - t)\log(1-t)).
        \end{align*}
    \end{proof}

\section{Mod-\texorpdfstring{$\phi$}{Phi}-convergence, moderate and large deviations} \label{section_mod_phi}
    In this section we study further stochastic properties of the random variables  $H_n(s, t)$  defined in \eqref{def_h}.
    We are particularly interested in the recently introduced concept of mod-$\phi$-convergence [see  \cite{femeni2016}]
    and large deviation properties.  The different limiting results for the sequence  $a_n^{-1} \big(H_n(s, t) - \e{H_n(s, t)}\big)$
    obtained in this section and in Section~\ref{section_applied_convergence} are summarised in Figure \ref{fig1}.

   To be precise consider the strip
        \begin{align*}
            S_{(a, b)} = \{z \in \mathbb{C} \mid a < \Re(z) < b\},
        \end{align*}
where  $a < 0 < b$ and let $\phi$ be a non-constant infinitely divisible distribution with moment generating function
$ \exp(\eta(z))$. Let  $(t_n)_{n \in \mathbb{N} } $  denote a real valued sequence and $(X_n)_{n \in \mathbb{N} } $  be a sequence
of real-valued random variables with existing  moment-generating functions on $S_{(a, b)}$
such that $t_n \to \infty$  and that for some non-vanishing analytic function $\psi$  on $S_{(a, b)}$
     \begin{align*}
            \exp(-t_n \eta(z)) \e{\exp(z X_n)} \to \psi(z)
        \end{align*}
        holds locally uniform on $S_{(a, b)}$.
        Following   \cite{femeni2016} the  sequence $(X_n)_{n \in \mathbb{N}}$  is said to converge mod-$\phi$ on $S_{(a, b)}$ with speed $(t_n)_{n \in \mathbb{N}}$.

     Mod-$\phi$-convergence is a very strong mode of convergence that implies the asymptotic behaviour of $(X_n)_{n \in \mathbb{N}}$ at different scales. Most prominently, Berry-Esseen bounds and large deviation results can be derived from mod-$\phi$-convergence. Particularly the large deviation results are stronger than the results that are usually obtained by a large deviation principle. The former gives an asymptotic equivalent for the probability $\p{X_n \ge t_n x}$, while the latter only yields the limiting behaviour for the logarithm of the probability.

   The core idea behind mod-$\phi$-convergence is that the distribution of $X_n$ is close to the distribution of the sum of $t_n$ i.i.d. $\phi$-distributed random variables. The function $\psi(z)$ measures the error made in this approximation and yields further refinement in the asymptotic formulas. \bigskip
    \begin{figure}[H]
        \small
        \begin{tikzpicture}[scale=.4]
            \usetikzlibrary{decorations.pathreplacing}
            \draw (0, 0) -- (0, 10);
            \draw [decorate,decoration={brace,amplitude=1em}] (-.3,0) -- (-.3, 7.5);
            \node at (-2.5, 3.25) [anchor=east, align=right]{Moderate Deviation \\ principle \emph{(Theorem~\ref{moderate_deviations})}};
            \draw [decorate,decoration={brace,amplitude=1em},] (-1,5) -- (-1, 10);
            \node at (-2.5, 8)  [anchor=east, align=right]{Relative error in \\  normal approximation \\ \emph{(Corollary~\ref{cormodphi})}};
            \draw (-.1, 10) -- (.1, 10) node[anchor=west]{0};
            \draw (-.1, 7.5) -- (.1, 7.5) node[anchor=west]{$n^{1/2}$};
            \node[anchor=west, align=left] at (4, 7.5) {Central limit Theorem \emph{(Theorem~\ref{convergence_hn})}  \\ Berry-Esseen bounds \emph{(Corollary~\ref{cormodphi})}};
            \draw (-.1, 5) -- (.1, 5) node[anchor=west]{$n^{5/9} p_n^{1/9}$};
            \draw (-.1, 2.5) -- (.1, 2.5) node[anchor=west]{$n^{2/3} p_n^{1/3}$};
            \node[anchor=west] at (4, 2.5) {Precise tail limits \emph{(Corollary~\ref{cormodphi})}};
            \draw (-.1, 0) -- (.1, 0) node[anchor=west]{$n p_n$};
            \node[anchor=west, align=left] at (4, 0) { Law of large numbers \emph{(Theorem~\ref{thm_lln})} \\ Large deviation principle \emph{(Theorem~\ref{large_deviation})}};
        \end{tikzpicture}
        \caption{ \textit{Overview of all the limiting results for the sequence $a_n^{-1} \big(H_n(s, t) - \e{H_n(s, t)}\big)$, depending on the order of $a_n$.}}
        \label{fig1}
    \end{figure}

    In the following we will first establish mod-$\phi$-convergence of the sequence   $(H_n (s,t))_{n\in \mathbb{N}}$ defined in (\ref{def_h}). Large deviation principles
    are discussed in the second part of this Section. As the moment generating function is closely related to cumulants, we first provide estimates for these objects
in Lemma~\ref{lemma_cumulant1}  and   Lemma~\ref{lemma_cumulant2}, which are proved in Appendix~\ref{secc}.

    \begin{lemma} \label{lemma_cumulant1}
    Let  $     r_{2n + 1, i, j} \sim \beta\left(\frac{p_n + 1}{2}(2n - i + 2) - j/2, j/2\right)$ denote independent beta distributed random variables, then
the cumulants of the random variables
        \begin{align*}
            S_n &= \sum \limits_{i = 1}^{\lfloor nt \rfloor - 1}\sum \limits_{j = 1}^{\lfloor p_ns \rfloor - 1}(\lfloor nt \rfloor - i)\log(r_{2n + 1, 2i, j}), \\
            S_n' &= \sum \limits_{i = 1}^{\lfloor nt \rfloor}\sum \limits_{j = 1}^{\lfloor p_ns \rfloor - 1}(\lfloor nt \rfloor - i + 1)\log(r_{2n + 1, 2i - 1, j})
        \end{align*}
        satisfy  the inequalities
        \begin{align*}
            \int_0^{t - 2/n} \Big (\frac{t - 2/n-x}{1 - x}\Big )^m \, dx &\le \frac{(-1)^{m}\kappa_m(S_n)}{n \frac{\lfloor p_ns -1 \rfloor\lfloor p_ns\rfloor}{(p_n + 1)^m} \frac{(m - 1)!}{4}} \le \Big(1 + \frac{m}{p_n}\Big ) \int_0^t \Big (\frac{t - x}{1 - x}\Big )^m \, dx \\
            \int_0^{t - 2/n} \Big (\frac{t - 1/n-x}{1 + 1/n - x}\Big )^m \, dx &\le \frac{(-1)^{m}\kappa_m^n(S_n')}{n \frac{\lfloor p_ns -1 \rfloor\lfloor p_ns\rfloor}{(p_n + 1)^m} \frac{(m - 1)!}{4}} \le \Big (1 + \frac{m}{p_n}\Big ) \Big (\int_0^t \Big (\frac{t - x}{1 - x}\Big )^m \, dx  + \frac{1}{n}\Big )
        \end{align*}
        for all $m \ge 1$. In particular
                \begin{align*}
            \max \{|\kappa_m(S_n)|, |\kappa_m(S_n')|\} \le 2 \cdot (m + 1)! n p_n^{2 - m}.
        \end{align*}
    \end{lemma}

    \begin{lemma}\label{lemma_cumulant2}
        Let $p_{2n + 1, i, j} \sim \beta\left(\frac{p_n + 1}{2}(2n - i + 2) - j/2, \frac{p_n + 1}{2}(2n - i + 2)\right)$ denote independent beta distributed random variables, then the cumulants of the random variables
        \begin{align*}
            T_n &= \sum \limits_{i = 1}^{\lfloor nt \rfloor}\sum \limits_{j = 0}^{\lfloor p_ns \rfloor - 1} \log(p_{2n + 1, 2i, j}^{\lfloor nt \rfloor - i + 1}(1 - p_{2n + 1, 2i, j})^{\lfloor nt \rfloor - i}), \\
            T_n' &= \sum \limits_{i = 1}^{\lfloor nt \rfloor}\sum \limits_{j = 0}^{\lfloor p_ns \rfloor - 1} \log(p_{2n + 1, 2i - 1, j}^{\lfloor nt \rfloor - i + 1}(1 - p_{2n + 1, 2i - 1, j})^{\lfloor nt \rfloor - i + 1})
        \end{align*}
        satisfy the inequalities
        \begin{align*}
            \max\{|\kappa_m(T_n)|, |\kappa_m(T_n')|\} &\le 6 \cdot 4^m (m + 1)! p_n^{-m} \left(n p_n + (\log(n) + 1)p_n^2\right) \\
            &\le 12 \cdot 4^m (m + 1)! n p_n^{2-m}
        \end{align*}
        for all $m \ge 1$.
    \end{lemma}

    \begin{thm}[Mod-Gaussian-convergence]\label{mod_phi_conv}
        For any fixed $s, t \in (0, 1]$ the sequence
        \begin{align*}
            \left(\frac{p_n}{n}\right)^{1/3}(H_n(s, t) - \e{H_n(s, t)})
        \end{align*}
        converges mod-Gaussian on any strip $S_{(a, b)}$ ($-\infty < a <  b < \infty $)
        with speed
            \begin{align}
            t_n = \left(\frac{p_n}{n}\right)^{2/3} \kappa_2(H_n(s, t))
            \sim{}& n^{1/3} p_n^{2/3} \frac{s^2}{2} \int_0^t \Big (\frac{t - x}{1 - x}\Big )^2 \, dx \to \infty \label{speed_tn}
        \end{align}
       and limiting function
        \begin{align*}
            \psi(z) = \exp\Big (-z^3 \frac{s^2}{6} \int_0^t \Big (\frac{t - x}{1 - x}\Big )^3 \, dx\Big).
        \end{align*}
    \end{thm}
    \begin{proof}
      Recalling the definition of $   H_n(s, t)$ in   (\ref{def_h}), it follows by an application of
     Lemma \ref{lemma_cumulant1} and \ref{lemma_cumulant2}    that $t_n \to \infty$. More precisely
        \begin{align*}
            t_n ={}& 2\frac{p_n^{2/3}}{n^{2/3}}  n \frac{(\lfloor p_n s\rfloor - 1) \lfloor p_n s\rfloor}{(p_n + 1)^2} \frac{1}{4} \Big (\int_0^t \Big (\frac{t - x}{1 - x}\Big)^2 \, dx + o(1)\Big ) + \frac{p_n^{2/3}}{n^{2/3}} O(n p_n^{-1} + \log(n))\nonumber \\
            \sim{}& n^{1/3} p_n^{2/3} \frac{s^2}{2} \int_0^t \Big (\frac{t - x}{1 - x}\Big )^2 \, dx \to \infty ,
        \end{align*}
        which proves \eqref{speed_tn}.
        By the definition of the cumulant-generating function the equation
        \begin{align*}
            &\log \Big\{\exp(-t_n z^2/2) \e{\exp(z p_n^{1/3} n^{-1/3} (H_n(s, t)) - \e{H_n(s, t)})}\Big\} \\
            ={}& \sum \limits_{j = 3}^\infty \frac{z^j}{j!} \kappa_j(p_n^{1/3} n^{-1/3}H_n(s, t)) \\
            ={}& \frac{z^3}{6} p_n n^{-1} \kappa_3(H_n(s, t)) + \sum \limits_{j = 4}^\infty \frac{z^j}{j!} p_n^{j/3} n^{-j/3} \kappa_j(H_n(s, t)).
        \end{align*}
        holds. As before we get by an application of Lemma \ref{lemma_cumulant1} and \ref{lemma_cumulant2}
        \begin{align*}
            \frac{z^3}{6} p_n n^{-1} \kappa_3(H_n(s, t))
            ={}& - \frac{z^3}{3} \Big (\int_0^t \Big (\frac{t - x}{1 - x}\Big)^3 \, dx + o(1)\Big ) \frac{p_n \lfloor p_n s -1 \rfloor \lfloor p_n s\rfloor}{(p_n + 1)^3} \frac{1}{2} \\
            & + z^3 O(p_n n^{-1} p_n^{-3} (n p_n + (\log(n) + 1) p_n^2)) \\
            ={}&-\frac{z^3}{6} s^2 \int_0^t \Big (\frac{t - x}{1 - x}\Big)^3 \, dx + o(1)
        \end{align*}
The remainder converges locally uniform to $0$, which follows from the inequality
        \begin{align*}
            \Big |\sum \limits_{j = 4}^\infty \frac{z^j}{j!} p_n^{j/3} n^{-j/3} \kappa_j(H_n(s, t))\Big |
            \le{}& 24 \sum \limits_{j = 4}^\infty \frac{|z|^j}{j!} p_n^{j/3}n^{-j/3} (j + 1)! 4^j n p_n^{2 - j} \\
            ={}& 9216 |z|^4 n^{-1/3} p_n^{-2/3}\sum \limits_{j = 0}^\infty (j + 4) \Big (\frac{4|z|}{n^{1/3} p_n^{2/3}}\Big )^j ~.
        \end{align*}
    \end{proof}
    From the mod-$\phi$-converge established in Theorem \ref{mod_phi_conv}  we can derive several conclusions. Exemplarily
    we mention the following results, which are obtained by an application of   Proposition 4.1.1, Theorem 4.3.1 and  Theorem 4.2.1 in \cite{femeni2016}
    \begin{cor}  \label{cormodphi}~~\\
    \vspace{-.5cm}
    \begin{itemize}
    \item[(1)]  Berry-Esseen bound
        \begin{align*}
            \sup \limits_{x \in \mathbb{R}} \Big|\mathbb{P} \Big ( {\frac{H_n(s, t) - \e{H_n(s, t)}}{\sqrt{\var{H_n(s, t)}}} \le x}  \Big ) - \Phi(x)\Big| = o\left(\frac{1}{\sqrt{t_n}}\right) = o\left(n^{-1/6} p_n^{-1/3}\right).
        \end{align*}
    \item[(2)] Relative error in normal approximation:
        For any sequence $x_n$ satisfying $x_n = o(t_n^{1/6}) = o(n^{1/18} p_n^{1/9})$ we have
        \begin{align*}
            \mathbb{P} \Big ( {\frac{H_n(s, t) - \e{H_n(s, t)}}{\sqrt{\var{H_n(s, t)}}} \le x_n}  \Big) = \Phi(x_n)(1 + o(1)).
        \end{align*}
      \item[(3)]  Precise tail limits:
        For $x > 0$ we have
        \begin{align*}
            &\mathbb{P} \Big ( {H_n(s, t) - \e{H_n(s, t)} \ge \frac{n^{1/3} t_n}{p_n^{1/3}} x } \Big)  \\
            ={}& \frac{1}{x \sqrt{2\pi t_n}} \exp\Big (-t_n x^2/2 -x^3 \frac{s^2}{6} \int_0^t \Big (\frac{t - y}{1 - y}\Big )^3 \, dy\Big)(1 + o(1)).
        \end{align*}
For $x < 0$ we have
        \begin{align*}
            &\mathbb{P} \Big ( {H_n(s, t) - \e{H_n(s, t)} \le \frac{n^{1/3} t_n}{p_n^{1/3}} x } \big )  \\
            ={}& \frac{1}{-x \sqrt{2\pi t_n}} \exp\Big (-t_n x^2/2 -x^3 \frac{s^2}{6} \int_0^t \Big (\frac{t - y}{1 - y}\Big )^3 \, dy\Big )(1 + o(1)).
        \end{align*}
    \end{itemize}
    \end{cor}
     In  Part (3)  of Corollary \ref{cormodphi}, the speed of the large deviation bound is $t_n \sim C n^{1/3} p_n^{2/3}$ (c.f. formula (\ref{speed_tn})) and $H_n(s, t)$ is rescaled by $\frac{n^{1/3} t_n}{p_n^{1/3}} \sim C n^{2/3} p_n^{1/3}$. This rescaling correponds to  a moderate deviation principle. Using the Gärtner-Ellis theorem we can strengthen the large deviation principle to a speed of $n p_n^2$ and a rescaling with $n p_n$ [see Theorem \ref{large_deviation} below]. Before we do this,
     we will provide a more general version of a moderate deviation principle for the sequence $H_n(s, t)$.
          This result contains part (3) of Corollary \ref{cormodphi} as a
           special case (using  $a_n = n^{2/3} p_n^{1/3}$). However, the  latter provides  more precise information about the limit.

    \begin{thm}[Moderate deviations] \label{moderate_deviations}
        Let $s, t \in (0, 1]$ be fixed and let $a_n$ be a sequence with  $\sqrt{n} = o(a_n)$ and $a_n = o(n p_n)$. Then
        \begin{align*}
            X_n = \frac{H_n(s, t) - \e{H_n(s, t)}}{a_n}
        \end{align*}
        satisfies a large deviation principle with speed $b_n = \frac{a_n^2}{n}$ and good rate function
        \begin{align*}
            I(x) = x^2 \Big (s^2 \int_0^t \Big (\frac{t - y}{1 - y}\Big )^2 \, dy\Big)^{-1}
        \end{align*}
    \end{thm}
    \begin{proof}
        We will use  the Gärtner-Ellis theorem [see Theorem 2.3.6 in \cite{demzeit2009} and  the subsequent remark].    From Lemma \ref{lemma_cumulant1} and Lemma \ref{lemma_cumulant2} it follows
        \begin{align*}
            \frac{1}{b_n}\log \e{\exp(b_n x X_n/a_n)} =& \sum \limits_{j = 2}^\infty \frac{x^j b_n^{j - 1}\kappa_j(X_n)}{j!a_n^j}
            ={} \frac{x^2}{2n} \kappa_2(X_j) + O\Big (\sum \limits_{j = 3}^\infty \frac{x^j a_n^{j - 2}}{j!} (j + 1)! \frac{n}{p_n^{j-2 }} \Big ) \\
            ={}& \frac{x^2}{2n} \Big (\int_0^t \Big (\frac{t - y}{1 - y}\Big )^2 \, dy + o(1)\Big ) n \frac{(\lfloor p_n s \rfloor - 1) \lfloor p_n s \rfloor}{(p_n + 1)^2} \frac{1}{2} \\
            &+ \frac{x^2}{2n} O\Big (\frac{n}{p_n} + (\log(n) + 1) \Big )
            + O\Big (\sum \limits_{j = 3}^\infty x^j (j + 1) \Big (\frac{a_n}{n p_n}\Big )^{j - 2}\Big) \\
            ={}& x^2 \frac{s^2}{4} \int_0^t \Big (\frac{t - y}{1 - y}\Big )^2 \, dy + o(1).
        \end{align*}
        Since $I(x)$ is the Fenchel-Legendre transform of the limit on the right hand side of the last equation,
                the Gärtner-Ellis theorem yields the desired result.
    \end{proof}

    The large deviation principle in Theorem \ref{moderate_deviations} is called a moderate deviation principle, because the scale at which the deviations occur is between the scaling of a central limit theorem and the scale of a law of large numbers. Indeed, at scale $\sqrt{n}$ the sequence $H_n(s, t) - \e{H_n(s, t)}$ is asymptotically Gaussian by Theorem \ref{convergence_hn} and at scale $n p_n$ the sequence $H_n(s, t)$ satisfies a law of large numbers by Theorem~\ref{thm_lln}.

    \begin{thm}[Large deviations]\label{large_deviation}
      The sequence $\frac{H_n(s, t)}{n p_n}$ satisfies a large deviation principle with speed $n p_n^2$ and good rate function
        \begin{align*}
            \Lambda^*(x) = \sup\limits_{\lambda > -\frac{1}{t}} \Big (\lambda x + \frac{s^2}{2} \int_0^t \log\Big  (1 + \lambda \frac{t - y}{1 - y}\Big  ) \, dy\Big ).
        \end{align*}
      In particular   the sequence $\frac{H_n(s, 1)}{n p_n}$ satisfies a large deviation principle with speed $n p_n^2$ and good rate function
        \begin{align*}
            \Lambda^*(x) =
            \begin{cases}
                -(x + s^2/2) + s^2 \log(s) + \frac{s^2}{2} \log(-2x) & x < 0 \\
                \infty & x \ge 0
            \end{cases}.
        \end{align*}
    \end{thm}
    \begin{proof}
      The Gärtner-Ellis theorem, Lemma \ref{lemma_cumulant1} and Lemma~\ref{lemma_cumulant2} yield
        \begin{align*}
            \Lambda(x) = & \limsup \limits_{n \to \infty} \frac{1}{n p_n^2} \log \e{\exp(p_n x H_n(s, t))}
            ={} \limsup \limits_{n \to \infty} \sum \limits_{j = 1}^\infty \frac{x^j}{j!} p_n^{j - 2} n^{-1} \kappa_j(H_n(s, t)) \\
            ={}& \limsup \limits_{n \to \infty} \sum \limits_{j = 1}^\infty\Big\{ \frac{1}{2} \frac{(-x)^j}{j} (s^2 + o(1)) \Big  (\int_0^t \Big  (\frac{t - y}{1 - y}\Big  )^j \, dy + o(1)\Big ) \\
            &\qquad \qquad +O\Big  (4^j x^j (j + 1) \Big  (\frac{1}{p_n} + \frac{\log(n) + 1}{n}\Big  )\Big  )\Big\} \\
            ={}& \frac{s^2}{2} \int_0^t \sum \limits_{j = 1}^\infty \Big (-x \frac{t - y}{1 - y}\Big  )^j \frac{1}{j} \, dy
            ={} -\frac{s^2}{2} \int_0^t \log\Big  (1 + x \frac{t - y}{1 - y}\Big  ) \, dy.
        \end{align*}
        The remaining part follows by a straightforward calculation of
      \begin{align*}
            \Lambda^*(x) = \sup \limits_{\lambda > -1}  \Big\{ \lambda x + \frac{s^2}{2} \log(1 + \lambda) \Big\}.
        \end{align*}
         \end{proof}

    \begin{rem}   {\rm
    Note that   the mod-Gaussian convergence provides  moderate deviation principles  while the application of the  Gärtner-Ellis theorem yields the full large deviation principle.
    This is most likely due to the fact that we used a rather simple approximation when proving Theorem \ref{mod_phi_conv}, based on  upper bounds for the cumulants
    of order larger than three.  In contrast to this, Theorem \ref{large_deviation} uses all the cumulants in calculating the limiting function.
        This situation is similar to Example 2.1.3 in \cite{femeni2016}, where  a sequence is proven to converge mod-Poisson as well as mod-Gaussian, and
         the results obtained from the mod-Poisson convergence are much stronger than the ones obtained from mod-Gaussian convergence. \\
        For $t = 1$ the limit of the cumulant generating function $\Lambda(x)$ in Theorem \ref{large_deviation} is $\Lambda(x) = -\frac{s^2}{2} \log(1 + x)$. This is the cumulant-generating function of a negative $\gamma(\tfrac{s^2}{2}, 1)$-distribution. A comparison with Theorem 4.2.1 from \cite{femeni2016} suggests the conjecture
         that $C \cdot (p_n + O(1))H_n(s, t)$ converges mod-$\gamma$ with a speed $t_n \sim C n p_n^2$, as this would imply the large deviation principle in Theorem \ref{large_deviation}.
        }
    \end{rem}

\bigskip

{\bf Acknowledgements.}
The work of H. Dette and D. Tomecki  was partially supported by the Deutsche
Forschungsgemeinschaft (DFG Research Unit 1735, DE 502/26-2, RTG 2131). The authors would like to thank Christoph Thäle for some helpful discussion about mod-$\phi$-convergence.

    \addcontentsline{toc}{section}{References}

redpajama_set_name: RedPajamaArXiv


Strike threats by employees at the state Malawi Broadcasting Corporation (MBC) compelled President Bingu wa Mutharika on Monday to summon chief executive officer Bright Malopa and director of programmes and chairperson of the MBC radio and TV merger Chimwemwe Banda to his private Ndata Palace in Thyolo.
Malopa, Banda and a third officer went to Ndata as irate employees gathered at the station waiting for them to get a convincing statement on the much long awaited merger of radio and TV. Mutharika is residing at the opulent Ndata mansion.
"I can confirm to you that the three have been summoned to Ndata House by the President in view of the strike on Tuesday," said one employee.
The employee said workers simply want to be told the truth about who is to remain working and who is to go and those going demand to have their benefits ready.
Malopa and his team returned and were said to be briefing the employees.
He was recently quoted in the local media that government has allocated funding for only 413 employees in the 2011-2012 national budget.
The other employees, he said will be re-deployed to other government departments but government sources said on Monday that it is doubtful if such employees will be re-deployed because those departments did not include them in the national budgets.
Most of the employees who have failed to be taken on board in the merged MBC do not have qualifications for entry into government.
MBC board chairperson was recently quoted in Weekend Nation that some of the employees are not deployable.
Over 20 employees told management that they simply want their money to leave for other pastures.
Some of such employees include Gerald Viola, Mbauwo Chavula, Jane Hauya and others.
Meanwhile, the planned August 23 strike has been averted following lengthy meeting with the MBC Board which has promised to resolve all concerns by 30th August.


text:
New Sculpture by Harold Grinspoon Debuts at Bay Path University
By BusinessWest Staff November 5, 2018 2111
LONGMEADOW — A new public sculpture, "LIFE," has been installed on the Bay Path University campus. The piece, designed and crafted from a fallen branch of a live oak tree, was created by philanthropist Harold Grinspoon, a longtime resident of Longmeadow and friend of the university.
A dedication for "LIFE" will be held on Wednesday, Nov. 7 at 3 p.m. in front of Elliot Hall, and is open to the public. In case of inclement weather, the event will be rescheduled. "LIFE" will be on view at Bay Path University for the next two years.
"We are honored to have LIFE featured so prominently on our campus," said President Carol Leary. "It is a perfect complement to the surroundings and creates a dramatic outdoor art experience for our students and the public. We are deeply grateful for Harold Grinspoon's generosity."
After successful careers in business and philanthropy, Grinspoon, at the age of 87, developed a new passion: art. His work as a sculptor started when a towering cherry tree fell in his backyard. Over the course of time, the fallen tree captured his imagination, only to take form and become a lasting piece of artwork. This first sculpture was eventually placed on view at the Mount, the Lenox home and museum of celebrated author Edith Wharton. It was the starting point for the ensuing 26 unique pieces of large-scale sculptures made from reclaimed trees, selected locally, as well as imported from Florida, California, and elsewhere.
Now at age 89, Grinspoon produces his sculptures in a bustling workshop in Agawam with a large team. His work is also currently included in three curated group exhibitions with SculptureNow at the Mount; XTCA: Cross Town Contemporary Art Exhibition presented by the University Museum of Contemporary Art at UMass Amherst; and Art in the Orchard in Easthampton. His works have also been displayed in three public locations: MGM Springfield, the Eastern States Exposition in West Springfield, and Mass General Hospital in Boston.
Brews, Bites & Bids to Benefit Big Brothers Big Sisters of Hampden County
Webber and Grinnell's Bailey Eastman Named MAIA 2018 Young Agent of the Year
PeoplesBank Announces Two Appointments
By BusinessWest Staff January 31, 2018
Bradley Users Advised to Plan Ahead and Allow Extra Time
By BusinessWest Staff November 20, 2018
Impact of Ballot Question 4 is Focus of ACCGS Lunch 'n' Learn
By BusinessWest Staff January 5, 2015

redpajama_set_name: RedPajamaCommonCrawl


Would you neglect to maintain your car? If you did, would you be surprised if it broke down? Would you neglect to maintain your car? If you did, would you be surprised if it broke down?
Then, why do you neglect routine maintenance and tune-ups for your computer??
Most people are accustomed to doing regular oil changes, brake pad maintenance and more for their cars and motorcycles – even for RVs, ATVs and other recreation vehicles. Most computer users think nothing of leaving their PC untouched for years. They then wonder why it or runs poorly or is slow.
The principles of routine maintenance should be applied to your PCs as well. Just running windows updates or downloading and installing updates for frequently used programs are not the same as proper maintenance.
Are you aware that can we perform a PC Tune-Up for you remotely?
Disks cleaning and reorganization are part of the maintenance we can perform remotely.
We check your backups are working properly, that disk space is adequate, that your anti-malware is not just updating, but that it's the latest version you should be running for your operating system.
We'll also look at your older programs and if they are no longer used, we'll recommend they are uninstalled.
Traditional hard drives can benefit from a defrag – so that large and frequently used files are where the operating system can find them quickly.
We have been helping customers with their PCs and offering advice on upgrade for many years, and we can help you!
Yes, let's get my PC Tuned-Up!


Booking.com: Hotels in Stalís. Book your hotel now!
Dias hotel and apartments are located in Stalida, in a picturesque setting suitable for families and couples.
Located in the seaside town of Stalida, 492 feet from the beach, Castello Apartments offer self-catering accommodations with free Wi-Fi. Within 328 feet, guests can find restaurants, bars and cafes.
Stalis Bay Apartments provides accommodations with a seasonal outdoor swimming pool and free WiFi, and is located in Stalís. There's also a fridge, stovetop and an electric tea pot.
Boasting a seafront location and unobstructed Cretan Sea views, Skajado Holiday Apartments offers elegantly decorated, self-catering accommodations, just a 5-minute walk from Stalida.
Set amid a Mediterranean garden full of colorful flowers, the family-run Psaras Apartments lies just 16 feet from the sandy beach of Stalis in Crete.
Just a few yards from the sandy Stalis Beach in Heraklion, 4S Beach Superior Apartments features a furnished sun terrace with Cretan Sea views.
Surrounded by hills, Latania Apartments provide free parking and affordable, large apartments and studios in Stalis, just 100 ft from the 2.5 mile long Stalis Beach.
On a beach in Stalís, this 5-star hotel features 3 freshwater pools and a poolside bar with a terrace. Recreational facilities include a Turkish bath, sauna and a fitness center.
Located in Stalís, Xenia Stalis Appartments offers accommodations with a balcony and free WiFi. A fridge and electric tea pot are also offered. Heraklio is 20 miles from the apartment.
Only 656 feet from Stalida Beach, in a peaceful environment, Everest Apartments features a spacious swimming pool and a snack bar.
Offering an outdoor pool and views of the garden, Villa Stella Apartments is located in Stalida in the Crete Region. Heraklio Town is 17 mi away. Free WiFi is offered in the rooms.
Just 98 feet from the sandy Stalis Beach, Aggello Boutique Hotel has a swimming pool and a poolside restaurant and snack bar, set within a palm-tree garden.
The family-run Lia Sofia Apartments provides self-catered accommodations in a privileged, quiet area, just 262 feet from the sandy Stalida Beach and a 5-minute walk from the center of Stalida.
Featuring an outdoor pool, a poolside bar and a sea-view terrace, Malliotakis Beach Hotel is located right on the sandy beach of Stalida. It offers rooms with a balcony or patio and free WiFi access.
W Suites is located in Stalida. Complimentary WiFi is featured. All units come with a terrace, a kitchenette with a fridge, and a private bathroom.
87 properties found in Stalís.


Home What Memories and Legacy Would You Leave Behind if You Died Today?
Have you ever stopped to wonder what people would say about you if you died? How many people would remember what you did, how you helped them, what you left behind?
Has that ever occurred to you?
Perhaps you are at the stage in your life where although you have accomplished everything you wanted, you are still feeling unfulfilled. Or possibly, you have a strong urge to do something different and you don't know what that 'something special' might be.
The truth of the matter is that we all look to develop ourselves, to fulfill the urge of contribution to individuals, to society, to the whole World.
Now you are in thinking mode, all your brain cells are moving a bit quicker, your breathing becomes faster and faster, your emotions starts to build up, you are overwhelmed with the desire to help others and without thinking much you take a pen and a piece of paper and start writing things you can do for others.
Some resonate with you and some don't, but it doesn't matter. Now you can't stop yourself but externalise all your internal feelings on the paper. Your emotions raise even more and your thoughts almost start to physicalise on the paper and it feels amazing. Now from your created list you choose the thing that resonates with you the most and you begin making a plan to make it happen.
You are at the peak of your excitement because you can almost see your plan already manifested, you see big groups of people gathering around and laughing, dancing and experiencing today as if they had never lived on this planet before, as if they never been through challenging times, as if they've never experienced pain.
As you look around you see yourself observing that group of people feeling peaceful, satisfied and content knowing that you have made a contribution to the lives of those people, that all of them are happy because of your actions. You are grateful to God, the Universe, Great Masters & Teachers, Supreme Beings of Light and all the people who made your dream of helping other people possible.
Now as your emotions becoming calmer your sense of peace, love and appreciation increases, you are standing up and taking the necessary steps – making those plans work for the benefits of a chosen person, a group of people and eventually the whole of humanity.
Does the thought of seeing people happy make you happy? Would you feel even greater if you could see the materialisation of your good deeds?
I believe that all of us came to this planet for a reason, and we all have different reasons, but there is only one main cause/reason that unites us – contribution. By being grateful for all the blessings we get every day and sharing with others we change the vibration on the planet therefore – we change the future.
There is a quote from Nelson Mandela that says "here are so many men and women who hold no distinctive positions but whose contribution towards the development of society has been enormous". Contribute to the world and your whole life will be filled with love, peace and fulfillment. And that's one of the main reasons for us to temporarily occupy our space on this beautiful planet Earth.


Shree Swaminarayan Gurukul is it your service all the time (24/7) Since 1948. We know service to you, is Service to God. We heartfully congratulate you all for believing us and joining your child in a Gurukul organization which makes him a complete human being, without any doubt. Your child is our child trust us.
You are not just a parent to your child but you are a family member of Shree Swaminarayan Gurukul organization. We always welcome you to be a part of us.
We surrender at the feet of Bhagwan Shree Swaminarayan for blessing us to do service to you all. We are even ready to render our service to manymore ahead. Please join us to feel the fragrance of Trust, Love, Care and Service for more details please visit us.
Shree Swaminarayan School is nothing but Perfect combination of Vidya, Sadvidya and Brahamavidya. The school discipline is up to the mark.
According to me this Shree Swaminarayan Gurukul School is not only good in academic but also the service which they provide is ultimate. I am really very happy that my child is learning in this such a great school.
Among the hundreds of schools we have in our region.


On THURSDAY 19th OCTOBER at 4pm, FOWS will be holding their AGM (Annual General Meeting) to report back on the charity's fundraising 2016-17 and to secure the positions of Chair, Vice Chair, Treasurer and Secretary for the forthcoming year.
A few parents have already voiced interest in taking up the key roles to work alongside all our FOWS members (parents and friends of Wrenbury School) in the year ahead, to continue raising much needed funds for the benefit of all the children at school.
The meeting is taking place, after the start of the film on Film NIght, in the school library.
Do come along; the more hands, heads and ideas we have will be all the better to make FOWS a success. We look forward to seeing you.


The air conditioning system is only as good as the contractor that installs it. We recommend you do your homework and research what you are looking for and that you hire a reputable contractor to install your air conditioning unit. We are license, bonded and insured.


text:
De Genevois (letterlijk het Geneefse) is een oude provincie van het hertogdom Savoye, in de omgeving van Genève. De stad Genève zelf hoorde er nochtans niet bij. Naast de hoofdstad Annecy zijn de voornaamste plaatsen: Thônes, Faverges en La Clusaz. De provincie werd begrensd door Carouge in het noordwesten, Faucigny in het noordoosten en door Savoye in het zuiden.
In de 11e eeuw bestond de provincie uit gebieden tussen het noorden van het meer van Bourget, de drempel van Faverges, en de valleien van de Aravis, de plateaus van de Bornes. De provincie omvatte ook bezittingen in Vaud en heerlijkheden in Grésivaudan.

Het werd samen met Savoye door Frankrijk geannexeerd in 1860; alhoewel het nooit een van de provincies van Frankrijk geweest is (die waren toen al afgeschaft), hoort het thuis in de lijst van voormalige provincies op het huidige Franse grondgebied.

Zie ook
graafschap Genève

Savoye
Historische provincie van Frankrijk
Natuurlijke regio in Frankrijk

redpajama_set_name: RedPajamaWikipedia


Sikkim Public Service Commission has announced to recruit 73 General Duty Medical Officers on Temporary Basis in Sikkim. Aspirants who have been searching for Government Jobs in Medical Sector can surly look forward to apply for these 73 Medical Vacancies in Sikkim. The official notification is consisting all the essential information regarding age limit, eligibility criteria n all. Those candidates who are having the Bachelor Degree of Medicine & Bachelor of Surgery will be eligible to apply.
73 General Duty Medical Officer Sikkim Exam consists written exam and interview. The written exam will consist of 3 papers. The selected candidates who will pass in written exam will be called for interview. Candidates need not submit any documents at the time of application. They should ensure that they are qualified on the date of application in all respects. Candidates will not be informed by the Sikkim Public Service Commission for any discrepancy or disqualification at personal level.
Bachelor of Medicine & Bachelor of Surgery from a recognized University. Should have completed Rotatory Internship of 01(one) year from a recognized Institute and Registered with Medical Council of India or Medical Council of any State.
Candidate must be in possession of either Sikkim Subject Certificate or Identification Certificate issued by the Competent Authority under relevant rules of the State Government.
Preference will be given to Sikkim Stipendiary seat reserved for State of Sikkim.
Candidates already in Government Service or in government owned undertaking or similar organization, whether in permanent or temporary capacity or as workcharged employee shall be, required to submit their application along with No Objection Certificate of their employer.
Candidates need not submit any documents at the time of application. They should ensure that they are qualified on the date of application in all respects.
Application form and syllabus can be downloaded from www.spscskm.gov.in The form should be filled up by the candidate himself/herself and submitted in the office of the Sikkim Public Service Commission on any working day between 10:30 a.m. to 03:00 p.m. along with original Bank Receipt of State Bank of Sikkim, for Rs. 150/- credited to '0051-SPSC'. The amount so deposited shall not be refunded or adjusted against any other purpose.
The complete filled in application form should reach the Secretary, Sikkim Public Service Commission, Old Tourism Office, M.G.Marg, Gangtok by hand or by post on or before 03:00 p.m. on 28/08/2015.
One STAMP SIZE photograph should be affixed in the application form in the space provided and an additional identical stamp size photograph with the candidates name on the back side of the photograph should be submitted along with the application form.


Look for a zillion trillion giveaways and reviews in the next week– I have a lot of bloggy catching up to do and there are some exciting prizes you'll want to be on the lookout for!
Happiest of birthdays to Roo! I hope she has a magical, cupcakey day!!!!!
well may roo have minny more realy nice b-days. tommrows my little ones b-day party but well she will not be 2 untell the 2nd.
Tell Veronica her cousin Brandon said Happy Birthday. And tell everybody I said hi.
Wow, the big 0-5! You go girl!
That's a beautiful picture of her! White really suits her well.


14 boys who have been part of the U14 County Development Squad were invited to Bromsgrove GC on Friday 1st June 2018 to take part in a County match. The format was to be 3 x 4BBB matches and a singles match between the two team captains, Elliot Jones (Gaudet Luce) and Sam Richmond (Bromsgrove).
All four matches were played in an excellent spirit, and the quality of the golf was very high. Many of the players will surely become the bedrock of future County Teams at U14, U16 and U18 level.
Pictured in their new County Caps, (back row l-r) George Stephens, Thomas Blizzard, Callum Brierley, Joshua Griffiths, Elliot Jones (Team Captain), Sam Richmond (Team Captain) & Tom Warbrick (Team Manager) (front row l-r) Nathan Wood, Sonny Priestley, Teddy Fee, Lewis Tolley, Josh Thorpe, Thomas Griffiths, Callum Jones & Joe Mead.
Whilst the match was won by Sam's Team 3 -1, the key reason for holding the fixture was to look at how the boys dealt with the whole day. In this respect, County U14 Team Manager & Coach, Tom Warbrick, was impressed by the boys attitude and how they all approached their matches. He can already see some potential foursomes partnerships for future County matches!
The course at Bromsgrove GC was in superb condition and the boys really enjoyed the whole day. The welcome from all the staff was second to none and nothing was too much trouble to ensure the lads had a day to remember. The County would highly recommend Bromsgrove GC for a visit.
The day was finished off by sandwiches and chips followed by ice cream, for tea. The two captains, Elliott and Sam, gave very articulate speeches, thanking Tom Warbrick and Bromsgrove GC for arranging and hosting the event.
The County is hoping to arrange another match for the U14 development squad later in the year - probably in the main school holidays.


When is the right time? Whenever you are ready. I would imagine that Dad would love to return to his spot in his bed. It would be great to try both kids together in a bed of their own; they might go for this if done together. Our children were used to sleeping in our bed at this age, but they eventually learned. Sometimes it took incentives, we started by making a "star-chart". For every night she fell asleep by herself she would get a star, and after three stars she would get a prize. Some nights she would say, "I don't think I want a star for tonight", and that was okay – she was always welcome in our bed. Soon, she was easily falling asleep by herself, but most mornings I would awaken to find her snuggling between my wife and I. Again, this was not a problem for us (I love snuggling my kids). So, the next step was to make a star chart for staying in her bed ALL NIGHT. She earned a really nice prize for that accomplishment.
Children develop a positive attitude about sleep.
They are not forced into anything that they aren't ready for – they get to choose between two nice things.
Everybody stays happy – there are no nighttime struggles, which help keep lost sleep to a minimum.
Kids are now older, sleeping great.


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