text:
\section{Introduction}
Multiresolution analysis of  deterministic signals by the use of wavelets has been extensively studied in recent years. However, in the context of stochastic processes, general wavelet approximations has not yet been fully investigated.  In the majority of cases developed deterministic methods and used error measures and metrics may not be appropriate to investigate wavelet representations of stochastic processes. It indicates the necessity of elaborating special stochastic techniques.

From a practical point of view, multi\-resolution analysis provides an efficient framework for the decomposition of random processes.
Wavelet representations could be used to convert the problem of analyzing a continuous-time random process to that of analyzing a random sequence, which is much simpler. This approach is widely used in statistics to estimate a curve given observations of the curve plus some noise, in time series analysis for smoothing functional data, in simulation studies of various functionals defined on realizations of a random process, etc.

Recently, a considerable attention was given to wavelet orthonormal series representations of stochastic processes. Numerous results, applications, and references on convergence of wavelet expansions of random processes in various spaces  can be found in \citealt{att, bar, cam, cla, did, kur, koz2, koz3, koz5, zha}, just to mention a few.

Figures~1 and 2\footnote{The figures have been generated by the R packages \textsc{geoR} and \textsc{wmtsa}. The Daubechies D8 wavelet basis and resolution levels 4 and 6 were used.} illustrate wavelet expansions of stochastic processes. A simulated realization of the process $\mathbf{X}(t)$ and its two wavelet reconstructions with different numbers of terms are plotted in Figure~1. Figure~2 displays boxplots of mean-square approximation errors for 500 simulated realizations of $\mathbf{X}(t)$ for each reconstruction.
Figure~2 suggests that empirical probabilities of large errors become smaller when the number of terms in the wavelet expansion increases.

Although the mentioned effect is well-known for deterministic functions, it has to be established theoretically for different stochastic processes and probability metrics. Numerical simulation results need to be confirmed by theoretical analysis. It is also important to obtain theoretical estimations of the rate of convergence for various stochastic wavelet expansions.

\begin{figure}[h]\label{fig1}
\begin{center}
  \includegraphics[width=0.7\textwidth]{Fig1.eps}
\end{center}
\hspace{2cm}\parbox{10cm}{\caption{Stochastic process and its two wavelet reconstructions}}
\end{figure}

Our focus in this paper is on convergence in $L_p([0,T])$ of wavelet expansions of $\varphi$-sub-Gaussian random processes. The paper extends the recent results on uniform convergence  in the papers \citealt{koz2,koz3,koz5}  to new classes of stochastic processes and probability metrics.

The analysis and the approach presented in the paper contribute to the investigations of wavelet expansions of random processes in the former literature. The approach is of a special interest if $p > 2,$ as it
extends the available $L_2$ results.
In that sense, Theorems~\ref{conv_Lp_Sub_ner}-\ref{str_Sub_covLp_nonstation} are
of special importance. The results are obtained under simple assumptions which can be easily verified.  The paper deals with the most general class of wavelet expansions in comparison with particular cases considered by different authors, see, for example, \citealt{cam,kur}.

The organization of the article is the following. In the second section we introduce the necessary background from the theory of $\varphi$-sub-Gaussian random variables and processes. Section 3 discusses wavelet expansions and approximations of stochastic processes.
In \S 4 we present the main results on convergence in $L_p([0,T])$ of  wavelet expansions of $\varphi$-sub-Gaussian random processes. In this section we also obtain the rate of convergence of the wavelet expansions and discuss some specifications for which the assumptions in the theorems can be easily verified.
\begin{figure}[t]\label{fig2}
\begin{center}
  \includegraphics[width=0.7\textwidth]{Fig2.eps}
\end{center}
\hspace{2cm}\parbox{10cm}{\caption{Boxplots of reconstruction errors for two approximations}}
\end{figure}
\section{$\varphi$-Sub-Gaussian random processes}
In this section, we review the definition of $\varphi$-sub-Gaussian random processes and  their relevant properties.

The space of $\varphi$-sub-Gaussian random variables  was first introduced in the paper
\citealt{koz4}.  More information about the space of
$\varphi$-sub-Gaussian random variables and processes can be found in
\citealt{bul,ant,koz0}.
\begin{defin} {\rm (\citealt{bul})} A continuous even convex function $\varphi(x),$ $x\in {\mathbb R},$ is called
an Orlicz N-function, if it is monotonically increasing for $x>0$,
$\varphi(0)=0,$ $ {{\varphi(x)}/{x}}\to 0,$ when $x\to 0,$ and $ {{\varphi(x)}/{x}}\to\infty,$  when $x\to\infty. $
\end{defin}

 \noindent{\bf Assumption Q}
Let $\varphi(\cdot)$ be an Orlicz N-function and
$\lim\limits_{x\rightarrow 0}{\varphi(x)}/{x^2}=c>0.$

\begin{rmk} The constant $c$ can be equal to $+\infty.$ \end{rmk}

\begin{prop} {\rm(\citealt{bul})} Every N-function $\varphi(\cdot)$ can be represented as
$\varphi(u)=\int_{0}^{|u|}f(v)\,dv,$ where $f(\cdot)$ is
a monotonically nondecreasing, right-continuous function, such that
$f(0)=0$ and $f(x)\to\infty,$ when  $x\to\infty.$ The function
$f(\cdot)$ is called a density of $\varphi(\cdot).$ \end{prop}

\begin{defin}{\rm (\citealt{bul})} Let $\varphi(x),x\in {\mathbb R},$ be an Orlicz N-function. The
function $\varphi^{*}(x),$ $x\in {\mathbb R},$ defined by the
formula $\varphi^{*}(x):=\sup_{y\in {\mathbb R}}(xy-\varphi(y))$ is called the
Young-Fenchel transform of $\varphi(\cdot).$ \end{defin}

Let $\{\Omega, \cal{B}, P\}$ be a standard probability space and $L_p(\Omega)$ denote a space of random variables having finite $p$-th absolute moments.

\begin{defin} {\rm(\citealt{ant})} Let $\varphi(\cdot)$ be an Orlicz
N-function satisfying the assumption $Q.$ A zero mean random
variable $\xi$ belongs to the space $Sub_\varphi(\Omega)$ (the space
of $\varphi$-sub-Gaussian random variables), if there exists a
constant $r_{\xi}\geq 0$ such that the inequality $\mathbf
E\exp\left(\lambda\xi\right)\le
\exp\left(\varphi(r_{\xi}\lambda)\right)$ holds for all $\lambda\in
{\mathbb R}.$ \end{defin}

\begin{prop} {\rm (\citealt{koz4})}
The space $Sub_\varphi(\Omega)$ is a Banach space with respect to
the norm $$ \tau_\varphi(\xi):=\inf\{a\ge0: \mathbf
E\exp\left(\lambda\xi\right)\le \exp(\varphi(a\lambda)),\lambda\in
{\mathbb R}\}.$$\end{prop}

\begin{rmk} A Gaussian centered random variable $\xi$ belongs to the space $Sub_\varphi(\Omega),$
where $\varphi(x)={x^2}/2$ and $\tau^2_\varphi(\xi)=\mathbf
E\xi^2.$
\end{rmk}

\begin{defin} {\rm (\citealt{bul})} Let $\mathbf{T}$ be a parametric space. A random process $\mathbf
X(t),$ $t \in \mathbf{T},$ belongs to the space $Sub_{\varphi}(\Omega)$ if $\mathbf X(t)\in Sub_{\varphi}(\Omega)$ for
all $t\in \mathbf{T}$  and $\sup_{t\in
\mathbf{T}}\tau_\varphi(\mathbf X(t))<\infty.$
\end{defin}

Let us denote $\tau_{\varphi}(t):=\tau_{\varphi}\left(\mathbf X(t)\right).$

\begin{defin} {\rm(\citealt{koz1})} A family $\Xi$ of random
variables $\xi\in Sub_{\varphi}(\Omega)$ is called strictly
$Sub_{\varphi}(\Omega)$ if there exists a constant $C_{\Xi}>0$
such that for any finite set $I,$ $\xi_{i}\in \Xi,$ $i\in I,$
and for arbitrary $\lambda_{i}\in {\mathbb R},$ $i\in I:$
\[\tau_{\varphi}\left(\sum\limits_{i \in
I}\lambda_{i}\xi_{i}\right)\leq
 C_{\Xi}\left(\mathbf E\left(\sum\limits_{i \in I}\lambda_{i}\xi_{i}\right)^2\right)^{1/2}.\]
$C_{\Xi}$ is called a determinative constant. \end{defin}

\begin{defin}\label{def_strictly Sub} {\rm(\citealt{koz1})} $\varphi$-sub-Gaussian random
process $\mathbf X(t),$ $t\in \mathbf{T},$ is called strictly
$Sub_{\varphi}(\Omega)$ if the family of random variables $\{\mathbf
X(t),t\in \mathbf{T}\}$ is strictly $Sub_{\varphi}(\Omega).$ The determinative
constant of this family is called a determinative constant of the
process and denoted by $C_{X}$.
\end{defin}

\begin{rmk} Gaussian centered random process $\mathbf X(t),$ $t\in \mathbf{T},$ is a strictly
$Sub_{\varphi}(\Omega)$ process, where $\varphi(x)={x^2}/2$ and the
determinative constant $C_{X}=1.$
\end{rmk}

\begin{thm}\label{ner_XSub_exp}{\rm(\citealt{koz0})}  Let $\{\mathbf{T},\Lambda,\mu\}$ be a measurable space and $\mathbf X(t),$ $t\in \mathbf{T},$
be a random process from the space $Sub_{\varphi}(\Omega).$

If $\int_\mathbf{T}
\left(\tau_{\varphi}(t)\right)^p\,d\mu(t)<\infty$ for some $p\ge 1,$ then the integral $\int_\mathbf{T}
\left|\mathbf X(t)\right|^p\,d\mu(t)$ exists with probability {\rm 1} and the following
inequality holds

\[\mathbf P\left\{\int_\mathbf{T}
\left|\mathbf
X(t)\right|^p\,d\mu(t)>\varepsilon\right\}\le 2\exp\left\{-\varphi^*\left(\left({\varepsilon}/c\right)^{1/p}\right)\right\},\]
for each non-negative
\begin{equation}\label{epsilon_ner}\varepsilon>c\cdot\left(f\left(p(c/\varepsilon)^{1/p}\right)\right)^p,\end{equation}
where $c:=\int_\mathbf{T} \left(\tau_{\varphi}(t)\right)^p\,d\mu(t)$
and $f(\cdot)$ is a density of $\varphi(\cdot).$

\end{thm}

\begin{expl} If $\varphi(x)={|x|^{\alpha}}/\alpha,$ $
1<\alpha\le2,$ then $f(x)=x^{\alpha-1}$ and
$\varphi^*(x)={|x|^{\beta}}/\beta,$ where $\beta>1$ and $1/{\alpha}+1/{\beta}=1.$ Hence,
$c\cdot\left(f\left(p(c/\varepsilon)^{1/p}\right)\right)^p=c^{\alpha}p^{\alpha-1}\varepsilon^{1-\alpha}.$
Therefore, inequality {\rm(\ref{epsilon_ner})} holds if
$\varepsilon>c\cdot p^{\frac{\alpha-1}{\alpha}p}.$
\end{expl}

\begin{rmk} If $\mathbf X(t),$ $t\in \mathbf{T},$ is a Gaussian centered random
process and $\sigma(t):=\left(\mathbf E\left(\mathbf
X(t)\right)^2\right)^{1/2},$ then inequality {\rm(\ref{epsilon_ner})} holds
true for $\varepsilon>\tilde c\cdot p^{\frac{p}{2}},$ where $\tilde
c:=\int_\mathbf{T} \left(\sigma(t)\right)^p\,d\mu(t).$ For such
$\varepsilon$ the following estimate is valid
\[\label{intX(t)gauss_ner}\mathbf P\left\{\int_\mathbf{T}
\left|\mathbf
X(t)\right|^p\,d\mu(t)>\varepsilon\right\}\le2\exp\left\{-\frac12\left(\frac{\varepsilon}{\tilde
c}\right)^{2/p}\right\}.\]
\end{rmk}

\begin{rmk} If $\mathbf X(t),$ $t\in \mathbf{T},$ is a strictly
$Sub_{\varphi}(\Omega)$ random process, then
\[c\le C^p_{X}\int_\mathbf{T}
\left(\mathbf E\left(\mathbf X(t)\right)^2\right)^{\frac
p2}\,d\mu(t).\]
\end{rmk}

\section{Wavelet representation of random processes}
In this section we introduce wavelet representations and approximations of non-random functions and stochastic processes.

Let $\phi(x),$ $x\in\mathbb R,$ be a function from the space
$L_2(\mathbb R)$ such that $\widehat{\phi}(0)\ne 0$ and
$\widehat{\phi}(y)$ is continuous at $0,$ where
$\widehat{\phi}(y)=\int_{\mathbb R}e^{-iyx}{\phi(x)}\,dx$ is the
Fourier transform of $\phi.$

 Suppose that the following assumption
holds true: $\sum_{k\in\mathbb Z} |\widehat{\phi}(y+2{\pi}k)|^2=1\
{\rm (a.e.)} $

There exists a function $m_0(x)\in L_2([0,2\pi])$, such that
$m_0(x)$ has the period $2\pi$ and
$$\widehat{\phi}(y)=m_0\left(y/2\right)\widehat{\phi}\left(y/2\right)\ {\rm (a.e.)}
$$
 In this case the  function $\phi(\cdot)$ is called
the $f$-wavelet.

Let $\psi(\cdot)$ be the inverse Fourier transform of the function
$$\widehat{\psi}(y):=\overline{m_0\left(\frac
y2+\pi\right)}\cdot\exp\left\{-i\frac
y2\right\}\cdot\widehat{\phi}\left(\frac y2\right).$$ Then the
function $\psi(x)=\frac1{2\pi}\int_{\mathbb
R}e^{iyx}{\widehat{\psi}(y)}\,dy$ is called the $m$-wavelet.

Let
\[\phi_{jk}(x):=2^{j/2}\phi(2^jx-k),\quad
\psi_{jk}(x):=2^{j/2}\psi(2^jx-k),\quad j,k \in\mathbb Z\,.
\]

It is known that the family of functions $\{\phi_{0k},
\psi_{jk}:j\in \mathbb N_0,\,k\in \mathbb Z\}$ is an orthonormal basis in
$L_2(\mathbb R)$ (see, for example, \citealt{dau}).

An arbitrary function $f(x)\in L_2(\mathbb R)$ can be represented in
the form
\begin{equation}\label{2.5}f(x)=\sum_{k\in\mathbb Z}\alpha_{0k}\phi_{0k}(x)+\sum_{j=0}^{\infty}\sum_{k\in\mathbb Z}\beta_{jk}\psi_{jk}(x)\,,
\end{equation}
$$\alpha_{0k}:=\int_{\mathbb R}f(x)\overline{\phi_{0k}(x)}\,dx,\quad \beta_{jk}:=\int_{\mathbb R}f(x)\overline{\psi_{jk}(x)}\,dx.$$
The representation (\ref{2.5}) is called a wavelet representation.

The series~(\ref{2.5}) converges in $L_2(\mathbb R)$ i.e.
$\sum_{k\in\mathbb
Z}|\alpha_{0k}|^2+\sum_{j=0}^{\infty}\sum_{k\in\mathbb
Z}|\beta_{jk}|^2<\infty\,.$

The integrals $\alpha_{0k}$ and $\beta_{jk}$ may also exist for
functions from $L_p(\mathbb R)$ and other function spaces. Therefore
it is possible to obtain the representation~(\ref{2.5}) for function
classes which are wider than $L_2(\mathbb R)$ (see, for example,
\citealt{jaf,tri}).

\noindent {\bf Assumption S} (\citealt{har})  For the
function $\phi(\cdot)$ there exists a decreasing function $\Phi(x),$ $x\ge
0,$ such that $\Phi(0)<\infty,$ $|\phi(x)|\le \Phi(|x|)$ (a.e.), and the integral
$\displaystyle\int_{\mathbb R}\Phi(|x|)\,dx$ is finite.

 Let
$\mathbf{X}(t),$ $t\in\mathbb R,$ be a  random process such that
$\mathbf E\mathbf{X}(t)=0\,$ for all $t\in\mathbb R$.
If sample trajectories of this process are  in the space
$L_2(\mathbb R)$ with probability one, then it is possible to obtain
the representation (wavelet representation)
\begin{equation}\label{X(t)}\mathbf{X}(t)=\sum_{k\in \mathbb Z}\xi_{0k}\phi_{0k}(t)+\sum_{j=0}^{\infty}\sum_{k\in \mathbb Z}
\eta_{jk}\psi_{jk}(t)\,, \end{equation} with mean-square integrals
\begin{equation}\label{intksieta}\xi_{0k}:=\int_{\mathbb R}\mathbf{X}(t)\overline{\phi_{0k}(t)}\,dt,\quad
\eta_{jk}:=\int_{\mathbb
R}\mathbf{X}(t)\overline{\psi_{jk}(t)}\,dt\,.\end{equation}

However, the majority of random processes does not possess the required
property. For example, sample paths of stationary processes are not
in the space $L_2(\mathbb R)$ (a.s.). However, in many cases it is
possible to construct a representation of type~(\ref{X(t)}) for
$\mathbf{X}(t).$

Consider the approximants of $\mathbf{X}(t)$ defined by
\[\label{x_nk}
\mathbf{X}_{n,\mathbf{k}_n}(t):=\sum_{|k|\le
k_0'}\xi_{0k}\phi_{0k}(t)+\sum_{j=0}^{n-1}\sum_{|k|\le
k_j}\eta_{jk}\psi_{jk}(t)\,,
\]
where $\mathbf{k}_n:=(k_0'(n),k_0(n),...,k_{n-1}(n)).$

For the sake of simplicity, we will omit the index $(n)$ in $(k_0'(n),k_0(n),...,k_{n-1}(n))$ and will use the notation $\mathbf{k}_n\to \infty$ to denote that  $n\to\infty,$ $k_0'\to\infty,$ and $k_j\to\infty$ for all
$j\in \mathbb N_0\,.$

Theorem~\ref{8103} below guarantees the mean-square convergence of
$\mathbf{X}_{n,\mathbf{k}_n}(t)$  to $\mathbf{X}(t).$

\begin{thm}\label{8103} {\rm (\citealt{koz5})} Let $\mathbf{X}(t),$ $t\in\mathbb R,$ be a random process
such that $\mathbf E\mathbf{X}(t)=0,$ $\mathbf
E|\mathbf{X}(t)|^2<\infty$ for all $t\in\mathbb R,$ and its
covariance function  $R(t,s)$  is continuous. Let the $f$-wavelet
$\phi$ and the $m$-wavelet $\psi$ be continuous functions and  the
assumption {\rm S}  hold true for both  $\phi$ and  $\psi.$  Suppose that there exist a function
$A:(0,\infty)\to (0,\infty)$ and $x_0\in\mathbb R$ such that  $
c(ax)\le c(x)\cdot A(a),$ for all $x\ge x_0.$

Assume that there exists an even non-decreasing  on $[0,\infty)$  function $c(x),$ $x\in\mathbb R,$
 with $c(0)>1$ such that $\int_{\mathbb R}c(x)\Phi(|x|)\,dx<\infty\quad {\rm and}
\quad |R(t,t)|^{1/2}\le c(t)$ for all $t\in\mathbb R.$ Then
\begin{enumerate}
  \item[\rm 1.]  $\mathbf{X}_{n,\mathbf{k}_n}(t)\in L_2(\Omega)\,;$
  \item[\rm 2.]  $\mathbf{X}_{n,\mathbf{k}_n}(t)\to \mathbf{X}(t)$ in mean square when  $\mathbf{k}_n\to\infty.$
\end{enumerate}
\end{thm}

\begin{rmk} It was shown  that the integrals in {\rm(\ref{intksieta})} exist if $\mathbf{X}(t)$ satisfies the assumptions of
Theorem~{\rm\ref{8103},} see {\rm \citealt{koz5}}.  Though there are other sufficient conditions in the literature which guarantee the existence of the integrals in {\rm(\ref{intksieta})} and the convergence in the mean-square sense in Theorem~{\rm\ref{8103},} the assumptions in terms of the functions $c(\cdot)$ and $\Phi(\cdot)$  can be easily verified in many practical examples.
\end{rmk}

\section{Convergence of wavelet expansions of
$\varphi$-sub-Gaussian random processes}

In this section we present the main results on convergence in $L_p(\mathbf{T}),$ $\mathbf{T}=[0,T],$ $T>0,$ of the wavelet expansions of $\varphi$-sub-Gaussian random processes. The rate of convergence in the space $L_p([0,T])$ is obtained. We also present some specifications of the general results for which the assumptions can be easily verified.

In \citealt{ant}, it was shown that there always exists a constant $c_{\varphi}>0$
such that $\mathbf E|\xi|^2\le c_{\varphi}\tau^2_{\varphi}(\xi).$ Therefore,  random processes from the space
$Sub_{\varphi}(\Omega)$ belong to the space
$L_{2}(\Omega).$ The following theorem is a corollary of  Theorem~\ref{8103} and the estimate $\left(R(t,t)\right)^{1/2}\le
\sqrt{c_{\varphi}}\tau_{\varphi}(t).$

\begin{thm}\label{meansq_subfi} Let $\mathbf{X}(t),$ $t\in\mathbb R,$ be a random process
such that $\mathbf{X}(\cdot)\in Sub_{\varphi}(\Omega)$ and $\mathbf{X}(t)$
is continuous in the norm $\tau_{\varphi}(\cdot).$ Let the functions
 $\phi(\cdot),$  $\psi(\cdot),$  $\Phi(\cdot),$ $A(\cdot),$ and $c(\cdot)$ satisfy the assumptions of Theorem~{\rm\ref{8103}}.

If $\int_{\mathbb R}c(x)\Phi(|x|)\,dx<\infty\quad {\rm and}
\quad \tau_{\varphi}(t)\le c(t)$ for all $t\in\mathbb R,$ then
\begin{enumerate}
  \item[\rm 1.]  $\mathbf{X}_{n,\mathbf{k}_n}(t)\in L_2(\Omega)\,;$
  \item[\rm 2.]  $\mathbf{X}_{n,\mathbf{k}_n}(t)\to \mathbf{X}(t)$ in mean square when  $\mathbf{k}_n\to\infty.$
\end{enumerate}
\end{thm}

\begin{defin} A random
process $\mathbf X(t),$ $t\in \mathbb{R},$ belongs to the space $L_p([0,T])$ if

\[\mathbf P \left\{\left(\int_0^T\left|\mathbf
X(t)\right|^p\,dt\right)^{1/p}<+\infty\right\}=1.\]
\end{defin}

\begin{thm}\label{conv_Lp_Sub_ner} Let $\mathbf{X}(t),$ $t\in\mathbb R,$ be a random process
such that $\mathbf{X}(\cdot)\in Sub_{\varphi}(\Omega)$ and $\mathbf{X}(t)$
is measurable and continuous in the norm $\tau_{\varphi}(\cdot).$
Let the assumptions of Theorem~{\rm\ref{meansq_subfi}} are satisfied.
Suppose that
\begin{equation}\label{inttau}\int_0^T\left(\tau_{\varphi}(t)\right)^p\,dt<\infty\end{equation} for
some $T>0$ and $p\ge 1.$

If
\begin{equation}\label{inttau_Xn-Xm}\int_0^T\left(\tau_{\varphi}\left(\mathbf X_{n,\mathbf k_n}(t)
-\mathbf X_{m,\mathbf k_m}(t)\right)\right)^p\,dt\to 0,\end{equation}
when $\mathbf k_n,\mathbf k_m\to\infty,$  then $\mathbf X_{n,\mathbf k_n}(t)\to\mathbf X(t)$ in the space $L_p([0,T])$ in
probability, when $\mathbf k_n\to\infty,$ i.e.  for all $\varepsilon>0$
\begin{eqnarray*}\label{P_int_Xn-X}\mathbf P \left\{\left(\int_0^T\left|\mathbf
X_{n,\mathbf k_n}(t) -\mathbf
X(t)\right|^p\,dt\right)^{1/p}>\varepsilon\right\}\to 0,\end{eqnarray*}
when $\mathbf k_n\to\infty$.

Furthermore, the following estimate
holds
\begin{equation}\label{P_int_Xn-Xm<exp}\mathbf P \left\{\int_0^T\left|\mathbf
X_{n,\mathbf k_n}(t) -\mathbf
X(t)\right|^p\,dt>\varepsilon\right\}\le2\exp\left\{-{\varphi}^*\left(\left({\varepsilon}/{c_n^{\infty}}
\right)^{1/p}\right)\right\},\end{equation} where
$c_n^{\infty}:=\int_0^T\left(\tau_{\varphi}\left(\mathbf
X_{n,\mathbf k_n}(t) -\mathbf X(t)\right)\right)^p\,dt$ and
$\varepsilon>c_n^{\infty}\left(f\left(p\,(c_n^{\infty}/\varepsilon)^{1/p}
\right)\right)^p.$
\end{thm}
\noindent{\it Proof.} First, we will show that the random variables $\xi_{0k}$
and $\eta_{jk}$ are from the space $Sub_{\varphi}(\Omega).$ We prove it only for $\eta_{jk}.$ The case of $\xi_{0k}$ can be dealt with similarly.

We will need the following generalization of the Minkowski inequality.
\begin{lem}\label{gmink}
If $\mathbf{X}(t),$ $t\in\mathbb R,$ is a measurable $\varphi$-sub-Gaussian random processes, then
\[\tau_{\varphi}\left(\int_{\mathbb R}\mathbf{X}(t)\,dt\right)\le\int_{\mathbb R}\tau_{\varphi}(t)\,dt.\]
\end{lem}
\noindent{\it Proof.} Without loss of generality we may assume that $\tau_{\varphi}(t)>0$ for all $t\in\mathbb R$  and $\int_{\mathbb R}\tau_{\varphi}(t)\,dt<\infty.$

Let us denote
\begin{eqnarray*}
 I: &=&  \mathbf E \left( \exp \left\{  \lambda
\int_{\mathbb R} \mathbf{X}(t)\,dt \cdot \left(
\int_{\mathbb R} \tau_{\varphi}(s)\,ds \right)^{-1}
\right\}\right)\\
&=& \mathbf E \left(\exp \left\{  \lambda
\int_{\mathbb R} \frac{\mathbf{X}(t)}{\tau_{\varphi}(t)} \cdot
\frac{\tau_{\varphi}(t)}{\int_{\mathbb R} \tau_{\varphi}(s)\,ds}\,dt
 \right\}\right),\
\lambda \in \mathbb R.
\end{eqnarray*}

Notice that $\exp(\cdot)$ is a convex function and
\[
  \int_{\mathbb R} \tau_{\varphi}(t) \cdot \left(
\int_{\mathbb R} \tau_{\varphi}(s)\,ds \right)^{-1} \,dt =1.
\]

Hence, by Jensen's inequality, we obtain
\[
\exp \left\{ \lambda
\int_{\mathbb R} \frac{\mathbf{X}(t)}{\tau_{\varphi}(t)} \cdot
\frac{\tau_{\varphi}(t)}{\int_{\mathbb R} \tau_{\varphi}(s)\,ds}\,dt
 \right\} \leq \int_{\mathbb R}  \exp \left\{ \lambda
\frac{\mathbf{X}(t)}{\tau_{\varphi}(t)}\right\} \cdot
\frac{\tau_{\varphi}(t)}{\int_{\mathbb R} \tau_{\varphi}(s)\,ds}\,dt.
\]

Therefore,
\begin{equation}\label{I}
I \leq \int_{\mathbb R} \mathbf E \left(\exp \left\{ \lambda
\frac{\mathbf{X}(t)}{\tau_{\varphi}(t)}\right\}\right) \cdot
\frac{\tau_{\varphi}(t)}{\int_{\mathbb R} \tau_{\varphi}(s)\,ds}\,dt\,.
\end{equation}

By the definition of $\tau_{\varphi}(\cdot)$ we get
\begin{equation}\label{est}
 \mathbf E\left( \exp \left\{ \lambda
\frac{\mathbf{X}(t)}{\tau_{\varphi}(t)}\right\}\right) \leq \exp \{ \varphi (\lambda) \}\,.
\end{equation}

By inequalities (\ref{I}) and (\ref{est}) we obtain $I\leq \exp \{ \varphi (\lambda) \}.$ Finally,  the statement of the lemma follows from the definition of $\tau_{\varphi}(\cdot).$
\hfill   $\Box$

By Lemma~\ref{gmink} we get
\begin{eqnarray*}\tau_{\varphi}(\eta_{jk})&\le&\int_{\mathbb R}\tau_{\varphi}(t)\left|\psi_{jk}(t)\right|\,dt\le
\int_{\mathbb R}\tau_{\varphi}(t)2^{j/2}\left|\psi(2^j
t-k)\right|\,dt\\
&\le& 2^{j/2} \int_{\mathbb R}c(t)|\psi(2^j t-k)|\,dt=2^{-j/2}\int_{\mathbb R}c\left(\frac{|u+k|}{2^j}\right)\psi(u)\,du\\
&\le& 2^{-j/2} \int_{\mathbb R}c\left(\frac{|u+k|}{2^j}\right)\Phi(|u|)\,du<+\infty,
\end{eqnarray*}
since $c(|u+k|)\le c(2u)\le c(u)A(2)$ for sufficiently large $u$.

Thus, the processes $\mathbf X_{n,\mathbf k_n}(t)$ belongs to the
space $Sub_{\varphi}(\Omega).$

To prove the theorem, it is enough to show that for all
$\varepsilon>0$
\begin{eqnarray}\label{P_int_Xn-Xm1}\mathbf P \left\{\int_0^T\left|\mathbf
X_{n,\mathbf k_n}(t) -\mathbf X_{m,\mathbf
k_m}(t)\right|^p\,dt>\varepsilon\right\}\to 0,\end{eqnarray} when
$\mathbf k_n$ and $\mathbf k_m\to+\infty.$

Indeed, by Theorem~\ref{ner_XSub_exp} and (\ref{inttau}), $\mathbf X(t)\in
L_p([0,T])$ with probability 1.

If the condition (\ref{P_int_Xn-Xm1}) is satisfied then
there exists a process $\mathbf Y(t)\in L_p([0,T])$ such that
\begin{eqnarray*}\mathbf P \left\{\int_0^T\left|\mathbf
X_{n,\mathbf k_n}(t) -\mathbf
Y(t)\right|^p\,dt>\varepsilon\right\}\to 0,\end{eqnarray*}
when
$\mathbf k_n\to+\infty,$
and by Theorem \ref{meansq_subfi} we get $\mathbf Y(t)=\mathbf X(t)$ with probability 1.

To prove (\ref{P_int_Xn-Xm1}) we denote
\begin{eqnarray*}c^m_n:=\int_0^T\left(\tau_{\varphi}\left(\mathbf X_{n,\mathbf k_n}(t)
-\mathbf X_{m,\mathbf k_m}(t)\right)\right)^p\,dt.\end{eqnarray*}
By Theorem \ref{ner_XSub_exp} the following inequality holds
\begin{equation}\label{intXn-Xm_ner_cnm}\mathbf P\left\{\int _T
\left|\mathbf X_{n,\mathbf k_n}(t) -\mathbf X_{m,\mathbf
k_m}(t)\right|^p\,dt>\varepsilon\right\}\le 2\exp\left\{-\varphi^*\left(\left({\varepsilon}/{c^m_n}\right)^{1/p}\right)\right\}\end{equation}
for
$\varepsilon>c^m_n\cdot\left(f\left(p\,(c^m_n/\varepsilon)^{1/p}\right)\right)^p.$

The condition (\ref{inttau_Xn-Xm}) implies  that
$c^m_n\to 0$ as $n,m\to\infty.$ The function $f(x)$ is
right-continuous and $f(0)=0.$ Hence
$f\left(p\,(c^m_n/\varepsilon)^{1/p}\right)\to 0, \ \
n,m\to\infty.$ Therefore, for arbitrary $\varepsilon>0$ and  sufficiently large
$n,m$  inequality (\ref{intXn-Xm_ner_cnm}) holds true. By definitions~1 and 2 the right-hand side of the inequality vanishes when $c^m_n\to 0,$ which implies  (\ref{P_int_Xn-Xm1}).

Notice, that $\mathbf X_{n,\mathbf k_n}(t) -\mathbf X_{m,\mathbf
k_m}(t)\to \mathbf X_{n,\mathbf k_n}(t) -\mathbf X(t)$ in $L_p([0,T])$ with probability 1 and $c^m_n\to c^\infty_n,$ when $\mathbf k_m\to+\infty.$
Hence, we obtain inequality (\ref{P_int_Xn-Xm<exp}) if we let $\mathbf k_m$ in (\ref{intXn-Xm_ner_cnm}) approach infinity.  \hfill   $\Box$

\begin{cor}\label{cor1_str_Sub}
Let  $\mathbf{X}(t),$ $t\in\mathbb R,$ be a strictly
$Sub_{\varphi}(\Omega)$ random process with a determinative constant $C_{X}.$ Then the
 condition {\rm(\ref{inttau})} holds true if  the integral
$\int_0^T\left(\mathbf E(\mathbf X(t))^2\right)^{p/2}\,dt$ is convergent.

If
\begin{equation}\label{int_Xn-Xm^2}\int_0^T\left(\mathbf E\left(\mathbf X_{n,\mathbf k_n}(t)
-\mathbf X_{m,\mathbf
k_m}(t)\right)^2\right)^{p/2}\,dt\to 0,\end{equation}
then the condition {\rm(\ref{inttau_Xn-Xm})} holds when $\mathbf k_n,\mathbf k_m\to\infty.$
The inequality {\rm(\ref{P_int_Xn-Xm<exp})} is valid for
\begin{equation}\label{c_n^infty}
c_n^{\infty}=C_{X}^p\int_0^T\left(\mathbf E\left(\mathbf
X_{n,\mathbf k_n}(t) -\mathbf
X(t)\right)^2\right)^{p/2}\,dt.\end{equation}
\end{cor}
\noindent{\it Proof.} The statement of  Corollary \ref{cor1_str_Sub}  follows from
 Theorem \ref{conv_Lp_Sub_ner}, Definition \ref{def_strictly Sub},
and the fact that a linear closure of a family of strictly
$Sub_{\varphi}(\Omega)$ random variables is a family of strictly
$Sub_{\varphi}(\Omega),$ see \citealt{koz1}. \hfill   $\Box$

\begin{cor}\label{cor2_str_Sub}
The statement of Corollary~{\rm\ref{cor1_str_Sub}} is valid if instead of  {\rm(\ref{int_Xn-Xm^2})} the  series
\[\sum_{k\in \mathbb Z}\left(\mathbf E\xi_{0k}^2\right)^{1/2}\left|\phi_{0k}(t)\right|
+\sum_{j=0}^{\infty}\sum_{k\in \mathbb Z}\left(\mathbf
E\eta_{jk}^2\right)^{1/2}\left|\psi_{jk}(t)\right| \]
converges uniformly on the interval $[0,T].$
Then the inequality {\rm(\ref{P_int_Xn-Xm<exp})} holds for
\begin{eqnarray}\label{c_n^infty2}
c_n^{\infty}=C_{X}^p\ T\, \cdot&&\hspace{-2mm}\sup\limits_{0\le t \le
T}\left(\sum_{|k|\ge
 k'_0}\left(\mathbf
E\xi_{0k}^2\right)^{1/2}\left|\phi_{0k}(t)\right|+\sum_{j=0}^{n-1}\sum_{|k|\ge
 k_j}\left(\mathbf
E\eta_{jk}^2\right)^{1/2}\left|\psi_{jk}(t)\right|\right.\nonumber\\
& &+\left.\sum_{j=n}^{\infty}\sum_{k\in
\mathbb Z}\left(\mathbf
E\eta_{jk}^2\right)^{1/2}\left|\psi_{jk}(t)\right|\right)^p.\nonumber\end{eqnarray}
\end{cor}
\noindent{\it Proof.}  The statement of Corollary \ref{cor2_str_Sub}  follows from (\ref{c_n^infty}) and
the estimate
$$\left(\mathbf E\left(\mathbf X_{n,\mathbf k_n}(t)
-\mathbf X(t)\right)^2\right)^{1/2}=\left(\mathbf
E\left(\sum_{|k|\ge
 k'_0}\xi_{0k}\phi_{0k}(t)+\sum_{j=0}^{n-1}\sum_{|k|\ge
  k_j}\eta_{jk}\psi_{jk}(t)\right.\right.$$
  $$\left.\left.+\sum_{j=n}^{\infty}\sum_{k\in
  \mathbb Z}\eta_{jk}\psi_{jk}(t)\right)^2\right)^{1/2}\le\sum_{|k|\ge
 k'_0}\left(\mathbf
E\xi_{0k}^2\right)^{1/2}\left|\phi_{0k}(t)\right|+\sum_{j=0}^{n-1}\sum_{|k|\ge
 k_j}\left(\mathbf
E\eta_{jk}^2\right)^{1/2}$$
  $$\times\left|\psi_{jk}(t)\right|+\sum_{j=n}^{\infty}\sum_{k\in
\mathbb Z}\left(\mathbf
E\eta_{jk}^2\right)^{1/2}\left|\psi_{jk}(t)\right|.$$
 \hfill   $\Box$
\begin{lem}\label{lem_ner_phi} If $\delta(\cdot)$ is an m-wavelet or an f-wavelet satisfying the assumption $S,$ then
\begin{equation}\label{sum|delta|}
\sup\limits_{x\in \mathbb R}\,\sum\limits_{k\in\mathbb Z}|\delta(x-k)|\le
3\Phi(0)+4\int_{1/2}^{\infty}\Phi(t)\,dt=:C_{\delta}
\end{equation}
and
\begin{equation}\label{supsum|delta|}\sup\limits_{|x|\le T}\sum\limits_{|k|\ge k_1}|\delta(x-k)|\le
\int_{k_1-T-1}^{\infty}\Phi(t)\,dt+\int_{k_1-1}^{\infty}\Phi(t)\,dt=:C_{\delta}(T,k_1)
\end{equation}
for $k_1\ge T+1.$
\end{lem}
\noindent{\it Proof.} Inequality (\ref{sum|delta|}) is a simple
modification of an inequality from \citealt{har}. Therefore we
only prove (\ref{supsum|delta|}).

By the assumption $S,$
$$\sum\limits_{|k|\ge k_1}|\delta(x-k)|\le \sum\limits_{k\ge k_1}\left(\Phi(|x+k|)
+\Phi(|x-k|)\right)=:z_{k_1}(x),$$ where $z_{k_1}(x)$ is an even
function. As $\Phi(\cdot)$ is a decreasing function on $[0,+\infty)$ and $k_1\ge T+1,$ we obtain
\begin{eqnarray*}\sup\limits_{|x|\le T}\sum\limits_{|k|\ge k_1}|\delta(x-k)|\le\sup\limits_{0\le x\le T}z_{k_1}(x)\le\sum\limits_{k\ge
k_1}\left(\Phi(|k-T|) +\Phi(|k|)\right)\\ \le\sum\limits_{k\ge k_1}
\left(\int_{k-1}^{k}\Phi(t-T)\,dt+\int_{k-1}^{k}\Phi(t)\,dt\right)\le
\int_{k_1-T-1}^{\infty}\Phi(t)\,dt+\int_{k_1-1}^{\infty}\Phi(t)\,dt.
\end{eqnarray*}
\hfill   $\Box$
\begin{thm}
Let $\mathbf{X}(t),$ $t\in\mathbb R,$ be a strictly
$Sub_{\varphi}(\Omega)$ random process for which the assumptions of
 Theorem~{\rm\ref{8103}} hold true and
$\int_0^T\left(\mathbf E(\mathbf X(t))^2\right)^{p/2}\,dt<+\infty.$

If
\begin{equation}\label{supsumksieta^2}\sup_{k\in \mathbb Z}\left(\mathbf
E|\xi_{0k}|^2\right)^{1/2}C_{\phi} +\sum_{j=0}^{\infty}\sup_{k\in
\mathbb Z}\left(\mathbf
E|\eta_{jk}|^2\right)^{1/2}\cdot 2^{j/2}C_{\psi}<\infty,\end{equation}
 then $\mathbf X_{n,\mathbf k_n}(t) \to\mathbf X(t)$ in the space $L_p(0,T)$ in probability, when
$\mathbf k_n\to\infty.$

Furthermore, for $\varepsilon>c_n^{\infty}\left(f\left(p\,(c_n^{\infty}/\varepsilon)^{1/p}
\right)\right)^p$  inequality
{\rm (\ref{P_int_Xn-Xm<exp})} is valid when
\begin{eqnarray}
c_n^{\infty}=C_{X}^p\ T&&\hspace{-1mm}\left(\sup_{k\in \mathbb Z}\left(\mathbf
E|\xi_{0k}|^2\right)^{1/2}C_{\phi}(T,k_0')+\sum_{j=0}^{J-1}\sup_{k\in \mathbb Z}\left(\mathbf
E|\eta_{jk}|^2\right)^{1/2}\right.\nonumber\\
&&\times\left.2^{j/2}C_{\psi}(T,k_j)+\sum_{j=J}^{\infty}\sup_{k\in
\mathbb Z}\left(\mathbf
E|\eta_{jk}|^2\right)^{1/2}\cdot2^{j/2}C_{\psi}\right)^p,\label{cn+}\end{eqnarray}
where $J:=\min\{n,\min\{j\in \mathbb{N}_0:\, k_j<2^jT+1\}\}.$
\end{thm}
\noindent{\it Proof.} Notice that $J\to \infty,$ when $\mathbf k_n\to\infty.$ By the choice of $J$ and (\ref{supsumksieta^2}) we get $c_n^\infty\to 0,$ when $\mathbf k_n\to\infty.$ The statement of the theorem  follows from
Theorem~\ref{conv_Lp_Sub_ner}, Corollary~\ref{cor2_str_Sub},
Lemma~\ref{lem_ner_phi}, and the fact $\sum_{k\in \mathbb
Z}\psi_{jk}(t)=2^{j/2}\sum_{k\in \mathbb Z}\psi(2^jt-k).$ \hfill   $\Box$

\begin{thm}\label{str_Sub_covLp4}
Let $\mathbf{X}(t),$ $t\in\mathbb R,$ be a strictly
$Sub_{\varphi}(\Omega)$ stationary  short-memory random process  with the covariance function $R(t-s):=\mathbf
E\mathbf{X}(t)\overline{\mathbf{X}(s)}.$ Let the assumptions of
 Theorem~{\rm\ref{8103}} are satisfied and
$\int_0^T\left(\mathbf E(\mathbf X(t))^2\right)^{p/2}\,dt<+\infty.$ Suppose that there exists  the Fourier transform $\widehat{R}(z)=\int_{\mathbb R}e^{-izt}{R(t)}dt$ and for some
$\alpha>0:$
$$\int_{\mathbb
R}\left|\widehat{R}(z)\right|\cdot\left|z\right|^{\alpha}\,dz<\infty.$$
If $\widehat{\psi}(\cdot)$ is a Lipschitz function of order
$\alpha/2,$  then {\rm(\ref{supsumksieta^2})} holds.
\end{thm}
\noindent{\it Proof.} By Parseval's theorem and the representation $\widehat{\psi}_{jk}(z)=2^{-\frac
j2}e^{-i \frac k{2^j} z}\widehat{\psi}\left(z/{2^j}\right),$ we deduce
$$\mathbf E|\eta_{jk}|^2=\left|\int_{\mathbb R}\int_{\mathbb R}R(u-v)\overline{\psi_{jk}(u)}\,du\,\psi_{jk}(v)\,dv\right|=\left|\int_{\mathbb R}\int_{\mathbb R}\frac{e^{-ivz}}{2 \pi}\widehat{R}(z)\overline{\widehat{\psi}_{jk}(z)}\,dz\psi_{jk}(v)\,dv\right|$$
\begin{equation}\label{st}\le\frac{1}{2 \pi}\int_{\mathbb R}\left|\widehat{R}(z)\right|\,\left|
\overline{\widehat{\psi}_{jk}(z)}\,\widehat{\psi}_{jk}(z)\right|\,dz=\frac{1}{2^{j+1}
\pi}\int_{\mathbb
R}\left|\widehat{R}(z)\right|\cdot\left|\widehat{\psi}\left(\frac
z{2^j}\right)\right|^2\,dz.
\end{equation}
By the Lipschitz conditions, (\ref{st}), and $\widehat{\psi}(0)=0,$ we obtain that  for all
$k\in\mathbb Z:$
\begin{equation}\label{st1}\left|\mathbf
E\eta_{jk}\right|^2\le\frac{C}{2^{j+1} \pi}\int_{\mathbb
R}\left|\widehat{R}(z)\right|\cdot\left(\frac
{|z|}{2^j}\right)^{\alpha}\,dz\le\frac{C}{\pi\,
2^{1+j(1+\alpha)}}\int_{\mathbb
R}\left|\widehat{R}(z)\right|\cdot\left|z\right|^{\alpha}\,dz,\end{equation}
where $C>0$ is the Lipschitz constant.

Similarly to (\ref{st}) we deduce
\begin{equation}\label{st2}\mathbf E|\xi_{0k}|^2\le \frac{1}{2\pi}\int_{\mathbb
R}\left|\widehat{R}(z)\right|\cdot\left|\widehat{\phi}\left(z\right)\right|^2\,dz<\infty\,.
\end{equation}
The integral in (\ref{st2}) is finite because the Assumption S implies $\phi(\cdot)\in L_1(\mathbb R)$ and therefore $\widehat{\phi}(\cdot)$ is bounded.

By  estimates (\ref{st1}) and (\ref{st2}) the series in (\ref{supsumksieta^2}) is convergent.
\hfill   $\Box$
\begin{thm}\label{str_Sub_covLp_nonstation}
Let $\mathbf{X}(t),$ $t\in\mathbb R,$ be a strictly
$Sub_{\varphi}(\Omega)$ random process
 with the continuous
covariance function $R(t,s):=\mathbf
E\mathbf{X}(t)\overline{\mathbf{X}(s)}.$  Suppose that
$\int_0^T\left(\mathbf E(\mathbf X(t))^2\right)^{p/2}\,dt<+\infty,$  there exist
$\widehat{R}_2(z,w):=\int_{\mathbb R}\int_{\mathbb
R}e^{-izu}e^{-iwu}{R(u,v)}\,dudv,$ and for some $\alpha>0:$
$$\int_{\mathbb
R}\int_{\mathbb
R}\left|\widehat{R}_2(z,w)\right|\cdot\left|z\right|^{\alpha}\cdot\left|w\right|^{\alpha}\,dzdw<\infty.$$
If  the assumptions of
 Theorem~{\rm\ref{8103}} are satisfied and $\widehat{\psi}(\cdot)$ is a Lipschitz function of order~$\alpha,$ then  {\rm (\ref{supsumksieta^2})} holds true.
\end{thm}
\noindent{\it Proof.} By Parseval's theorem,
\begin{eqnarray}
\mathbf
E|\eta_{jk}|^2&=&\int_{\mathbb R}\int_{\mathbb R}\mathbf E
\mathbf{X}(u)\overline{\mathbf{X}(v)}\
\overline{\psi_{jk}(u)}\psi_{jk}(v)\,dudv\nonumber\\
&=&\frac{1}{(2
\pi)^2}\int_{\mathbb R}\int_{\mathbb
R}\widehat{R}_2(z,w)\,\overline{\widehat{\psi}_{jk}(z)}\,\widehat{\psi}_{jk}(w)\,dz\,dw\nonumber\\
&\le&\frac{1}{2^{j}(2 \pi)^2}\int_{\mathbb R}\int_{\mathbb
R}\left|\widehat{R}_2(z,w)\right|\cdot\left|\widehat{\psi}\left(\frac
z{2^j}\right)\right|\cdot\left|\widehat{\psi}\left(\frac
w{2^j}\right)\right|\,dz\,dw.\label{st1_2}
\end{eqnarray}

 By properties of the $m$-wavelet
$\psi(\cdot)$ we have $\widehat{\psi}(0)=0.$ Therefore, using the Lipschitz
conditions, we obtain
$$\mathbf
E|\eta_{jk}|^2\le\frac{C^2}{(2 \pi)^2 2^{j(1+2\alpha)}}\int_{\mathbb
R}\int_{\mathbb
R}\left|\widehat{R}_2(z,w)\right|\cdot\left|z\right|^{\alpha}\cdot\left|w\right|^{\alpha}\,dz\,dw,$$
where $C>0$ is the Lipschitz constant.

Analogously to (\ref{st1_2})  we deduce that
\[\mathbf E|\xi_{0k}|^2\le
\frac{1}{4 \pi^2}\int_{\mathbb R}\int_{\mathbb
R}\left|\widehat{R}_2(z,w)\right|\cdot\left|\widehat{\phi}\left(
z\right)\right|\cdot\left|\widehat{\phi}\left(
w\right)\right|\,dz\,dw<\infty
\]
due to the boundedness of $\widehat{\phi}(\cdot).$

An  application of the above estimates gives the convergence of (\ref{supsumksieta^2}).
 \hfill   $\Box$

 \begin{rmk}  Conditions of Theorems {\rm\ref{str_Sub_covLp4}} and
  {\rm\ref{str_Sub_covLp_nonstation}} on the random process $\mathbf{X}(t)$ are formulated in terms of its spectral density. These conditions are related to the behavior of the high-frequency part of the spectrum. Such assumptions are standard in the convergence studies of stochastic approximations.
 \end{rmk}
\begin{rmk} If the assumptions of Theorem {\rm\ref{str_Sub_covLp4}} or
{\rm\ref{str_Sub_covLp_nonstation}} are satisfied, then
 {\rm(\ref{P_int_Xn-Xm<exp})} holds true for $c_n^\infty$ given by {\rm(\ref{cn+})}. All the terms in {\rm(\ref{cn+})} can be easily computed in practice for specific stochastic processes and wavelet bases.
\end{rmk}

\section{Conclusions}
The obtained results may have various practical applications for the approximation and simulation of random processes. The analysis of the rate of convergence provides a constructive algorithm for determining the number of terms in the wavelet expansions to ensure the approximation of stochastic processes with given accuracy.

The developed methodology and results are important extensions of the recent findings in the wavelet approximation theory of stochastic processes to the space $L_p([0,T])$ and the class of $\varphi$-sub-Gaussian random processes. This class plays an important role in generalizations of various theoretical properties of Gaussian processes.
In addition to classical applications of $\varphi$-sub-Gaussian random processes in signal processing, the results can also be used in new areas, like compressed sensing and actuarial modelling, consult, for example,  \citealt{lab,ver,yam}.

It would be of interest
\begin{itemize}
\item to adopt and specify the results for different wavelet bases, which satisfy the assumptions of the theorems, for example, Daubechies, Battle-Lemarie and Meyer wavelets;
\item to examine the tightness of the estimates by simulations.
\end{itemize}

redpajama_set_name: RedPajamaArXiv


text:
(1899–1958), historien de l'art norvégien, directeur du personnel gouvernemental, conjoint de Maja Refsum
  (1913–1976), acteur et chanteur
 Asbjørn Nilssen (1875-1958), coureur norvégien du combiné nordique et médecin norvégien
 Bjarne Nilssen (1879-1956), sportif et ingénieur norvégien
 Jappe Nilssen (1870-1931), journaliste et critique d'art norvégien du Dagbladet, à la fois écrivain et historien d'art
  (1958-), réalisateur norvégien et producteur de cinéma
  (1871–1947), parlementaire norvégien
 Olaug Nilssen (1977-), écrivaine norvégienne et auteure de littérature d'enfance
  (1829–1912), parlementaire norvégien, ordre de Saint-Olaf
  (1974-), compositeur norvégien et musicien de jazz, Buddyprisen
 Sebastian Plur Nilssen, musher
  (1990-), acteur norvégien

redpajama_set_name: RedPajamaWikipedia


text:
Indian Army creates Guinness record with 58 personnel riding ONE motorbike
The group of 58 corps that was led by Major Bunny Sharma were dressed in tricolour on the vanguard, while the bike was steered by Subedar Rampal Yadav.
In a spectacular display of daredevilry, a team of Army Service Corps (ASC) entered the Guinness Book of World Records. These 58 Indian army personnel rode a 500 cc Royal Enfield motorbike for 1,200m at the Yelahanka Air Force Station, Karnataka. Popularly known as Tornadoes, these army men set a new record at the Indian Air Force base in Bengaluru.
The group of 58 corps that was led by Major Bunny Sharma were dressed in tricolour on the vanguard, while the bike was steered by Subedar Rampal Yadav. The other people were standing on the flanks of the two wheels. Apparently, the ASC team hold 19 world and national records and have showcased their masterclass stunts at over 1,000 shows across the world.
Additional Directorate General of Public Information tweeted the record-breaking video saying, "Motorcycle team of #ArmyServiceCorps, #TORNADOES, broke another World Record on November 19, 2017 wherein 58 men rode on a single moving motorcycle.The act was performed for the Guinness Book of World Records, Limca Book of Records and Unique World Records at Yelahanka #Bengaluru".
Source by:-indianexpress
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A friend whose wife is a school teacher has been poking some fun at homeschoolers on facebook, and I had replied to a couple of his threads. He then asked me directly why we decided to do it, and I thought I might post my response here. This blog exists in part to organize my thoughts and in part to keep my memories of our lives. As I wrote that answer to my friend, I realized my memory of how we came to homeschool was already becoming fuzzy and it spurred me to get it out here before I forget entirely. So here, in short, is why we have become homeschoolers.
Oh, I just realized I didn't answer the "why". In retrospect it feels like a complicated thing to answer.
Our five year old is really excited about/distracted by action around him. He's bright and when he's interested in something he's borderline happily obsessive. The combination can be frustrating when a teacher has a big group of kids to work with. We learned this about him in preschool and at church.
Also, as we approached fall and had a decision to make about school (his birthday being in January meant it was optional to start this year), I started researching the neighborhood schools. They aren't horrible, not great, they have large classes and aren't strong in reading. I had a couple of friends who homeschool and started looking into that, too. At first I thought I'd just make sure he got a good reading start.
Mixing "gamillas"- vanilla banana cookies.
I also read a book called "How to Destroy the Imagination of Your Child", and "The Well Trained Mind". Those began to really tip me toward sincerely considering it. That was a shock even to me.
We went to an info meeting about a virtual public school but much of their system really is just like public, including grades from teachers and needing to complete subjects in grade level lock-step. Also, they wanted us to log 6 to 8 hours a day, which with one-on-one attention is just dumb in kindergarten. My impression was that it is mainly for people having trouble in public school.
Then I went to a huge homeschool conference here in portland. I could not believe the diversity of families there. Everything from head-covering mennonites to freaky tattoed people (that's us). There were so many transracial families, too. I remember being struck by that. Last month when we had our social worker interviews for the adoption she also mentioned how huge adoption is in the homeschool community. I think that's awesome. It was a beautiful thing to see.
Anyway, sorry to ramble but it was a curvy road. I love it now that we're doing it. We're in a "Classical Conversations" group and that is incredible. Very broad.
This is Toby's group, memorizing the history sentence: The war of 1812 gave confidence to the US to write the Monroe Doctrine, warning Europeans not to attempt to colonize the Americas.
After writing this response to my friend, I looked around for some good statistics on homeschool graduates. The statistics are overwhelmingly positive.


text:
ZME Travel
Indian area
Transportation, Western Europe
How to reach the city center from London's airports – Stansted
January 16, 2013 April 1, 2014 Tiberius Puiu
London is one of the busiest tourist attraction in Europe attracting some 30 million visitors every year, most which travel by plane to the UK capital. There's no wonder then that there are so many airports in the vicinity of the city – London Heathrow, London Stansted, London Gatwick, London Luton and London City.
With this in mind, it's rather easy to become distracted and confused on how to reach the actual city of London, especially considering all of the airports are tens of miles away. Today we'll be discussing the easiest routes to reach the city center by train, cab and public transit from the London Stansted airport.
London Stansted Airport (STN) is the UK's third busiest airport and one of the fastest growing airports in Europe, however it's also one of the farthest from London, located at some 64km (40 miles) away. The terminal has become especially attractive in the past decade since a lot of low cost airlines have made it their destination of choice.
1 To and from Stansted by rail
2 To and from Stansted by coach
3 Alternative transportation to and from Stansted
To and from Stansted by rail
The quickest way to leave the airport for Central London is definitely the Stansted Express, which runs every 15 minutes (30 minutes for the earliest and latest trains on Fridays and Mondays) and takes only 45 minutes to reach iverpool Street Station. Tickets cost £21.50 single and £29.50 for return. From here there are of course plenty of connections for the London Underground.
Here's the time table: 6am to 12.30am Monday to Thursday, and until 1.30am on Friday and Saturday. Sunday trains run from 5.30am until 1.30am.
To and from Stansted by coach
Conveniently, the closest bus station from Stansted is located opposite the main terminal entrance and provides regular services on a daily basis.
National Express operates a 24 hour service to Victoria Coach Station with stops at Golder's Green, Finchley Road Underground station, St John's Wood, Marylebone, Baker Street, Marble Arch and Hyde Park Corner. Journey time is 1 hour and 40 minutes. This service runs at intervals of up to an hour. Standard tickets cost £10 single/£17 return.
Terravision offers a low cost coach service into Central London from Stansted Airport. Terravision runs a non-stop service to Victoria Coach Station and a route to Liverpool Street Station. Standard fares are £9 single and £14 return.
Alternative transportation to and from Stansted
Cabs are very easy to find and pick from, with a number of companies being stationed in front of the airport – no black cabs though. The pricing varies, however all cabs have it listed on the meter so be sure to check it before hoping in – be prepared to dig into your pocket though. You should always ask what the fair would be to your destination Make no mistake, however, hitching a ride to London by cab is by far the most expensive option.
The least expensive option is not surprisingly the longest, but maybe the funnest – that is if you can handle peddling. Cyclists benefit from the use of a free bike shelter in the Short Stay Car Park, next to the terminal building, which is easy to get to and is one of the most straightforward ways to get in and out of the airport.
However, most people travel with luggage and quite a few usually. So another excellent opportunity is to get to Center London by car.
Many passengers choose to book Stansted airport parking and travel by car. There are numerous ways to do this – you can park in an on-site car park by booking direct with the airport, or you can book an off-site parking space through one of the many local car parking suppliers based near the airport. If you're hoping to keep the costs down, this is one of the cheapest ways to park near the airport – but it's not always the most convenient, as you will usually need to take a shuttle bus to and from the terminal.
If you're looking for airport parking deals, the best bet is to book well in advance and shop around for the best deals. Many off-site operators and those providing Meet and Greet services will often advertise deals and discounts, so it usually pays to spend some time researching as many different options as possible.
Tags: englandLondonUnited Kingdom
January 16, 2013 April 1, 2014 0 7,438
By Tiberius Puiu
A travel enthusiast keen on sharing musings from all the places, culture and people I've encountered thus far. Feel free to e-mail me for a bit of chat.
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Why you shouldn't miss out on Reykjavik's hot pools
A day in Birmingham for Nature Lovers: Harborne, Edgbaston Reservoir, and Sandwell
Architecture and art
Tintern Abbey – The hidden jewel in Wales
Nottingham, one of England's best-kept secrets
How to use GPS with Google Maps
Things to do for free in Kaunas
If Sea Levels Rose, Which Countries Would go First?
25 Breathtaking Bird's Eye View Pictures of the World
Get a sense of the metropolitan life in Shanghai
© 2014 ZME Travel. All Rights Reserved.


The Preface to The Picture of Dorian Gray is a Manifesto where the writer expounds his theory about writing, putting into discussion existing theories.
It consists of a collection of epigrams that aptly sums up the philosophical tenets of the artistic and philosophical movement known as Aestheticism, according to which "art is a beautiful object". Aestheticism, which found its footing in Europe in the early nineteenth century, proposed that art need not serve moral, political, or otherwise didactic ends.
The preface offers one of Wilde's most famous aphorisms: "There is no such thing as a moral or an immoral book. Books are well written, or badly written.
That is all." According to Wilde, the artist might consider the moral or immoral lives of people as part of the subject matter of a work, but art itself is not meant to instruct the reader. The true artist is not out to prove anything and makes no judgments of right or wrong. What people call "vices" or "virtues" are merely materials for the artist. Those who attempt to go beneath the surface of a work, or to read meaning into a symbol, do so at their own risk. Considerable disagreement about a work of art only proves that the work is "new, complex, and vital."
Oscar Wilde, The Picture of Dorian Gray, Chapter 1 – Basil's studio. The extract in exam opens with a detailed description of the setting. Here the writer gives voice to a series of sensorial images (visual, olfactory and auditive), thanks to which the reader can make a mental picture of the room where the indoor scene takes place. The connotation of the space is conveyed thanks to a careful, and obsessive too, focus on the furniture's details, typical of Aesthetic movement and Pre-Raphaelite art, which creates an exotic atmosphere of Persian taste ("the divan of Persian saddle-bags") where research for beauty is the order if the day.


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Update: To Celebrate 20 Eisner Nominations, Image is having a 50% Digital Sale!
EISNERS ANNOUNCE 20 NOMINATIONS FOR IMAGE COMICS
And the nominees are… On sale, 50% off all digital editions for a limited time
Comic-Con International announced the nominations for the Will Eisner Comic Industry Awards 2014 this morning, 17 of which were published by Image Comics. An additional three nominations were shared with other publishers, bringing the total nominations for Image Comics to 20.
Brian K. Vaughan and Fiona Staples' SAGA, a winner of previous Eisner awards, was nominated in four categories. The hit series was nominated for Best Continuing Series, writer Brian K. Vaughan was nominated for Best Writer, and artist Fiona Staples was nominated in the Best Painter/Multimedia Artist and Best Cover Artist categories.
Eric Stephenson and Nate Bellegarde's NOWHERE MEN held its own with three nominations and an additional shared nomination. The series was nominated in the Best Continuing Series category, writer and Image Comics Publisher Eric Stephenson was nominated for Best Writer, artist Nate Bellegarde was nominated for Best Penciller/Inker, and Jordie Bellaire was nominated in the Best Coloring category for several projects that included NOWHERE MEN, as well as other Image titles like THE MANHATTAN PROJECTS, PRETTY DEADLY, and ZERO.
Jonathan Hickman and Nick Dragotta's EAST OF WEST also raked in three nominations, including Best Continuing Series. Artist Nick Dragotta was nominated for Best Penciller/Inker and Frank Martin was nominated for Best Coloring for their work on the series.
Matt Fraction and Chip Zdarsky's SEX CRIMINALS, which has taken the comics industry by storm and was hailed by TIME Magazine as "comic series of the year," earned two nominations, one for Best Continuing Series and one for Best New Series. In addition, writer Matt Fraction was nominated for Best Writer for several series, including SEX CRIMINALS.
Kelly Sue DeConnick and Emma Ríos's PRETTY DEADLY received two nominations, plus another two shared nominations. Emma Ríos was nominated for Best Penciller/Inker and Best Cover Artist. The shared nominations included Kelly Sue DeConnick for Best Writer and Best Coloring for Jordie Bellaire.
Greg Rucka and Michael Lark's critically acclaimed dystopian action series LAZARUS was nominated in the Best New Series category, as was Shadowline/Image's fantasy series RAT QUEENS by Kurtis J. Wiebe and Roc Upchurch.
OUTLAW TERRITORY, VOL. 3, edited by Michael Woods, was nominated for Best Anthology.
"I couldn't be more proud of all the insanely awesome men and women currently working with us at Image Comics," said Eric Stephenson, Publisher at Image Comics. "One of the great joys of my life has been watching all of these incredible comics make the transition from an exciting idea to an amazing series, and I think our strong showing on this year's Eisner ballot only further underscores the strength and vitality of creator-owned comics."
In celebration of the nominated series, Image Comics will offer 50% off all the digital editions of the following nominated titles—available on the Image Comics Website immediately, and on ComiXology, Google Play, Amazon, and Apple iBooks beginning 4/16—through midnight EST on April 21, 2014: EAST OF WEST, NOWHERE MEN, SAGA, SEX CRIMINALS, LAZARUS, RAT QUEENS, PRETTY DEADLY, MANHATTAN PROJECTS, ZERO, and OUTLAW TERRITORY, VOL. 3.
Voting for the awards will open soon at www.eisnervote.com. All professionals in the comic book industry are eligible to vote. The deadline for voting is June 13. The results of the voting will be announced in a gala awards ceremony on the evening of Friday, July 25 at Comic-Con International in San Diego.
For more, check out the official Image Comics website.
FULL LIST OF IMAGE COMICS EISNER 2014 NOMINATIONS:
Best Continuing Series:
East of West, by Jonathan Hickman and Nick Dragotta
Sex Criminals, by Matt Fraction and Chip Zdarsky
Nowhere Men, by Eric Stephenson and Nate Bellegarde
Saga, by Brian K. Vaughan and Fiona Staples
Best New Series:
Lazarus, by Greg Rucka and Michael Lark
Rat Queens, by Kurtis J. Wiebe and Roc Upchurch (Image/Shadowline)
Best Anthology:
Outlaw Territory, edited by Michael Woods
Best Writer:
Eric Stephenson, Nowhere Men
Brian K. Vaughan, Saga
Best Penciller/Inker or Penciller/Inker Team:
Nate Bellegarde, Nowhere Men
Nick Dragotta, East of West
Emma Ríos, Pretty Deadly
Best Painter/Multimedia Artist (interior art):
Fiona Staples, Saga
Best Cover Artist:
Emma Rios, Pretty Deadly
Best Coloring:
Frank Martin, East of West
SHARED NOMINATIONS:
Kelly Sue DeConnick, Pretty Deadly
Matt Fraction, Sex Criminals
Jordie Bellaire, The Manhattan Projects, Nowhere Men, Pretty Deadly, Zero
See the complete list of 2014 Eisner Nominations right here.
ABOUT THE EISNER AWARDS
Named for acclaimed comics creator the Will Eisner, the awards are celebrating their 26th year of highlighting the best publications and creators in comics and graphic novels.
ABOUT IMAGE COMICS
Image Comics is a comic book and graphic novel publisher founded in 1992 by a collective of best-selling artists. Image has since gone on to become one of the largest comics publishers in the United States. Image currently has five partners: Robert Kirkman, Erik Larsen, Todd McFarlane, Marc Silvestri and Jim Valentino. It consists of five major houses: Todd McFarlane Productions, Top Cow Productions, Shadowline, Skybound and Image Central. Image publishes comics and graphic novels in nearly every genre, sub-genre, and style imaginable. It offers science fiction, fantasy, romance, horror, crime fiction, historical fiction, humor and more by the finest artists and writers working in the medium today. For more information, visit www.imagecomics.com.
Tags201450% SalecomixologyDigitalDigital SaleEisnerEisner NominationsImage ComicsNominationsNomineesSale
Complete 2014 Eisner Nomination List Released
X-O Manowar #24
Tyler Goulet
Tyler Goulet, owner/founder and editor-in-chief of All-Comic.com, has an addiction to comics that seemingly cannot be sated. Reading everything from Hellboy (his favorite character, series and creator) to Batman to Pluto, and other Manga, and most everything in between, there's not much he can't talk about or won't read. He currently resides in Calgary, Canada with his wife Kayla, his son Finan, a black cat named Hailstorm (yes, like the Transformer) and their always-starving, three legged idiot cat Lemmy. (yes, from Motorhead)
Death Vigil #7
Brubaker and Phillips blast into THE FADE OUT this August
Fans Can't Get Enough of THE FIX
Image/Top Cow launches Holy-warrior epic in MAGDALENA


This modern house for sale in Marbella, Nueva Andalucía is located next to the renowned golf course of Las Brisas. Only 5 minutes from the exciting Puerto Banús with its luxury shopping, exquisite restaurants and yachts. The property sits in an extremely private destination at the end of a cul de sac in a safe private community with 24/7 security offering superb open views towards the golf course and La Concha Mountain.
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Who was Achilles?
The Greek hero Achilles is one of the most famous figures in Greek myth and a key character in the Trojan War. Discover the story of this hero, from his infamous anger to his 'Achilles heel'.
British Museum 15 October 2019
Who were Achilles' parents?
Achilles was the son of Peleus, a Greek king, and Thetis, a sea nymph or goddess.
Zeus, the king of the gods and Poseidon, god of the sea, had both fallen in love with Thetis and were rivals for her hand in marriage. However, the gods were warned of a prophecy that Thetis would have a son who would grow up to be greater than his father. Worried by this, Zeus arranged for Thetis to marry a mortal man so that her child couldn't challenge his power. In another version of the story, Thetis rejects Zeus's advances and a furious Zeus decrees that she will never marry a god. Either way, Thetis ends up married to the mortal Peleus and Achilles is born.
Terracotta relief showing Peleus and Thetis, c. 490–470 BC. Thetis tries to resist marriage to Peleus by transforming her body into powerful elements such as fire and wild beasts, here a lion.
What's the story of his 'Achilles heel'?
Thetis gives birth to Achilles who, unlike her, is mortal. She attempts to make the baby Achilles immortal, by dipping him in the River Styx (the river that runs through the underworld), while holding him by his heel. The one part of his body left untouched by the waters becomes his only point of weakness, hence the phrase 'Achilles heel'.
Thomas Banks RA, Thetis and Achilles, 1789. © Victoria and Albert Museum, London.
In the Iliad there is no mention of Achilles' invulnerability or the weakness of his heel. In scenes representing his death in Greek art, he has been shown with an arrow in his torso, as well as his heel. This may indicate that the 'Achilles heel' story is a later addition to the myth.
The story reveals the sadness of a mother who knows that, being immortal, she will outlive her son.
Why was Achilles raised by a centaur?
Achilles' mother Thetis abandons her husband and son to return to live with the sea nymphs when Achilles is still young. Needing some help raising Achilles, Peleus sends him to be educated by a centaur named Chiron.
Centaurs, who have the upper body of a human and the lower body and legs of a horse, are often represented in Greek art as violent and savage creatures, but Chiron was known for his wisdom and had educated other heroes including Heracles and Jason.
Achilles instructed by Chiron in the Management of the Javelin. Print after Giovanni Battista Cipriani, 1789.
Under the care of Chiron, Achilles is fed a diet that includes the innards of lions and wild pigs, and the marrow of she-wolves, to make him strong. Chiron teaches the young Achilles hunting as well as music and intellectual pursuits.
Eugène Delacroix, The Education of Achilles, c. 1862. Pastel on paper.
Getty Center, Los Angeles.
Did Achilles have a male lover?
As a boy, Achilles develops a close relationship with another boy named Patroclus, who joins Achilles' household as an exile, having accidentally killed another child. They become friends and possibly lovers. In Homer's Iliad, Achilles describes Patroclus as 'the man I loved beyond all other comrades, loved as my own life'.
Drinking cup showing Achilles bandaging Patroclus' wound. c. 500 BC.
©ANTIKENSAMMLUNG, STAATLICHE MUSEEN ZU BERLIN
-PREUSSISCHER KULTURBESITZ- Photo: Johannes Laurentius.
Although Homer doesn't explicitly cast them as lovers, their intimate relationship is crucial to the plot of the Iliad, and in other works of ancient literature the relationship between the two men is more explicitly referenced as a love affair. In ancient Greece it was common for men to have sexual relationships with both men and women. There were no words in ancient Greek for 'homosexual' and 'heterosexual' and in classical Athens relationships between older, often married, men and younger men were a normal part of social life.
Why did Achilles' parents try to stop him going to Troy?
A prophecy had foretold that Achilles would die at Troy, so when the Greek leaders gather for their campaign to return the abducted Helen, Achilles' parents attempt to prevent him joining the expedition. He is hidden on the island of Skyros, disguised as a girl at the court of King Lycomedes among his numerous daughters.
Roman sarcophagus relief, depicting Achilles (centre) holding a helmet among the daughters of Lycomedes AD 150–200.
However, due to another prophecy, the Greeks know they can only win the war with Achilles' help. The Greek kings Odysseus and Diomedes discover his whereabouts and trick him into revealing himself so he can join the troops on the expedition to Troy. Disguised as travelling salesmen, they visit the court with jewellery and women's clothes. Amongst these goods they place weapons, which Achilles instinctively grabs and is found out.
Achilles wearing a dress as a disguise unsheaths a sword and is recognised. Etching after Peter Paul Rubens, 1630–1645.
What did Achilles do in the Trojan War?
Achilles arrives at Troy with 50 ships. He is the leader of the army known as the Myrmidons and is the best fighter on the side of the Greeks.
Storage jar showing Ajax (left) and Achilles (right) playing a game. 530–520 BC.
Troy is a well-defended city and nine long years of siege follow. The epic poem which covered this part of the war (the Cypria) does not survive, so its events are known in much less detail. In art, a popular scene was that of Achilles playing a board game with the hero Ajax. The image suggests that the Greek heroes spent many long hours whiling away the time during the siege of Troy. Things only really get interesting during the 10th year of the war, which is the focus for Homer's Iliad.
What caused 'the rage of Achilles'?
The anger or rage of Achilles is a key theme of Homer's Iliad. In fact 'anger' is the first word of the whole poem.
Achilles is initially angry because the leader of the Greek forces, King Agamemnon, takes a captive woman named Briseis from him. Early Greek society was highly competitive and a man's honour was vital to his sense of identity and position. By taking away the prize of honour that has been allocated to Achilles in recognition of his fighting prowess, Agamemnon dishonours him.
Drinking cup showing Briseis being taken from Achilles, who is depicted sulking with his cloak wrapped around him. c. 480 BC.
Achilles withdraws from battle and refuses to fight. When the Trojans make gains in the battle, Agamemnon agrees to send an embassy to Achilles to try to persuade him to re-join the fighting by offering him a wealth of gifts. Achilles remains unmoved but does allow his friend Patroclus to lead the Myrmidons into battle wearing Achilles' armour.
Henry Fuseli (1741–1825), Achilles Lamenting for Patroclus. Drawing, 1770.
Patroclus is killed in the bloody fighting by the Trojan prince Hector, who mistakes him for Achilles, and the real Achilles is utterly distraught. His mother, Thetis, consoles him and Achilles declares his intention to kill Hector and avenge the death of Patroclus, despite his weeping mother's revelation that, according to fate, his own death will soon follow that of Hector.
How does Achilles avenge Patroclus' death?
Achilles is determined to avenge Patroclus' death at any cost and announces that he is ending his anger against Agamemnon and will re-join the fighting. The two sides meet in battle and Hector waits outside the city gates, ready to fight Achilles.
Detail of a wine mixing bowl showing Achilles fighting Hector. c. 490 BC.
Achilles chases Hector around the wall of the city, then the two heroes fight, and Achilles plunges his spear into Hector's neck. As he lies dying, Hector appeals to Achilles to return his body for cremation – a request that is heartlessly refused – and with his last breath he prophesies Achilles' own death at the hands of the Trojan prince Paris.
Pietro Testa (1611–1650), Achilles dragging the body of Hector around the walls of Troy. Etching, 1648–50.
Achilles, with his lust for revenge still not satisfied, deliberately mistreats the body of Hector, tying him to his chariot and dragging him behind in the dirt as he drives back to the Greek camp.
Why does Achilles return the body of Hector?
Following the funeral of Patroclus, Achilles' grief makes him restless. He ties Hector's body to his chariot and repeatedly drags it around the tomb of Patroclus, in his furious need for retribution. However the gods protect Hector's body so that in spite of this cruel treatment it remains unblemished.
Storage jar depicting Achilles dragging the body of Hector behind his chariot. c. 520–500 BC.
Eventually Hector's father Priam, with the assistance of the god Hermes, makes the dangerous journey to the Greek camp to see Achilles and beg for the return of his son's corpse for burial. Their emotional encounter is powerfully depicted on this silver cup, which shows Priam coming to Achilles and kissing his hands.
I have endured what no one on earth has ever done before – I put to my lips the hands of the man who killed my son.
Iliad 24.505–6
Wedgwood plaque showing Priam begging Achilles (far right) for the return of Hector's body. Late 18th century.
The two men weep together and share a meal. In a transformation of character, which restores humanity to the hero and a sense of order to the world, Achilles agrees to release Hector's body. As dawn breaks, Priam returns to Troy with Hector's body and the funeral marks the end of the Iliad.
What happens to Achilles after the end of the Iliad?
Although the Iliad doesn't explore what happens to Achilles next, there are numerous later legends and other ancient authors tell more of his story. After the death of Hector, the Trojans, with their best fighter dead, call on their allies to help them defeat the Greeks. The Ethiopian King Memnon brings his army to support the Trojans, but is killed by Achilles in battle. Achilles also faces the Amazons – the tribe of female warriors – and fights their leader, Queen Penthesilea. At the moment Achilles kills her with his spear, their eyes meet and he falls in love with her, too late.
Storage jar showing Achilles killing the Amazon queen, Penthesilea. c. 530 BC.
How does Achilles die?
Achilles is killed by an arrow, shot by the Trojan prince Paris. In most versions of the story, the god Apollo is said to have guided the arrow into his vulnerable spot, his heel. In one version of the myth Achilles is scaling the walls of Troy and about to sack the city when he is shot. In other accounts he is marrying the Trojan princess Polyxena and supposedly negotiating an end to the war when Paris fires the shot that kills him.
Engraved cornelian scarab depicting wounded Achilles. Etruscan. c. 400–350 BC.
After his death, Achilles is cremated, and his ashes are mixed with those of his dear friend Patroclus. The Odyssey describes a huge tomb of Achilles on the beach at Troy, and Odysseus meets Achilles during his visit to the underworld, among a group of dead heroes.
How has Achilles been reinterpreted throughout history?
The complexity of Achilles' character has meant that he has been reinterpreted and reinvented throughout history.
For the ancient Greeks he was an archetypal hero who embodied the human condition. Despite his greatness he was still mortal and fated to die. A hero cult for Achilles developed in several areas across Greece where he was venerated and worshipped like a god. A structure known as the 'Tomb of Achilles' in the area where the city of Troy once stood became a place of pilgrimage for many in antiquity, including Alexander the Great.
Simon François Ravenet I (1706–1774), after Filippo Lauri, Alexander Visiting the Tomb of Achilles. Etching and engraving, 1769.
For the Romans, Achilles was on the one hand a model of military prowess but also, for poets such as Horace and Catullus, an archetype of brutality. By the medieval period, Achilles provided a model of how not to behave. Changes to the narrative in the markedly pro-Trojan versions of the myth that were dominant at this time made Achilles into a cowardly scoundrel who destroyed himself through his lustful passions. In Dante's Inferno, Achilles appears in the second circle of hell, that of lust.
Filippo Albacini (1777–1858), The Wounded Achilles. Marble, 1825. © The Devonshire Collections, Chatsworth. Reproduced by permission of Chatsworth Settlement Trustees.
In the Renaissance, when there was a renewal of interest in the classical world accompanying the reintroduction of Greek texts into western Europe, Achilles regained interest as a more complex character. By the early 19th century, the period of Romanticism, he was the perfect hero, embodying a life given over to emotion, and beauty doomed to ruin. A neoclassical sculpture of the period, The Wounded Achilles, shows the perfection of his body even in his dying moments. Achilles has also served as a heroic justification for the sacrifice of soldiers as well as a symbol of the destruction and brutality of war.
Max Slevogt (1868–1932), Achilles Frightens the Trojans. Print, 1905.
Achilles' complex character continues to be explored. Pat Barker's recent novel The Silence of the Girls focuses on the experience of his captive woman Briseis, and The Song of Achilles, by Madeline Miller, is written from the perspective of his lover Patroclus.
Achilles may be a killing machine but he is nevertheless deeply human and that is, perhaps, why his story is still compelling after more than 3,000 years.
Discover more about the story of Achilles in our major BP exhibition Troy: myth and reality, which runs from 21 November 2019 – 8 March 2020.


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1963 VW Beetle shipped from ID to CA
Jim and Debra Sklenar purchased a 1963 VW Beetle and needed it transported from Boise, ID, to their home in Santa Cruz, CA. Having never shipped a car before, they went on an internet search to find an auto hauling company. Looking for a transporter instead of a transport broker, they came across J&S Transportation
"We did an Internet search for car haulers. We were of course inundated with a number of emails from brokers, so just discounted them right off the bat," said Jim. "We were interested in a car hauler. My wife called J&S and spoke to the dispatch office. J&S said they had their own trucks but they did contract out for routes that they were not able to cover themselves and I was okay with that."
"We had a very positive response and had all our questions were answered. We decided we would use J&S regarding shipping a '63 Volkswagen Bug from Boise, ID, to Santa Cruz, CA."
The transport went very well. It was a very straightforward experience. We were told we could expect delivery of the car in about one week from the day that we called. We were in contact with the driver of the transporter all the way. They gave us plenty of notice of when the car would arrive, and sure enough it arrived seven days exactly from the day of we made the initial call.
"One concern that I had was that the car would be dropped at some location and then picked up again by another transport. If the actual shipping would have been in more than one part I was concerned that the car might languish at some mid way location before it was picked up by another carrier and brought down here the rest of the way. But that did not happen at all. The car showed up the day it was supposed to with no problem at all."
"This was our first time transporting a car and we much appreciate the positive experience. The whole interaction with J&S was very positive and at a very reasonable price, too. We got quotes for a little bit higher cost and J&S was right in the middle of the pack. We were very satisfied all the way around."
"What most impressed about J&S was the customer service we received. It was very good. We were in contact with the actual driver of the transporter; he contacted us several times along the route and telling us where he was and when he would be arriving, and was very easy to arrange a place for the vehicle of that size to park and unload. The customer service was a big plus."
"I would absolutely recommend them for all the reasons I mentioned. It was an overall positive experience and I would not hesitate to use J&S again if the occasion should arise."
Thanks, Jim. We'll be here if you need to ship a car in the future.
By Gerald S | April 17th, 2015 |
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redpajama_set_name: RedPajamaGithub


NMPRS Enter / Twenter show huge success for Stal Zegwaard!
The first foals of 2013 are born!
Beautiful pictures of three of our approved stallions in the snow. Above: Star's To Hot To Tango AMHA, bottom left Falabella Don Paco and bottom right Star's Toetanchamon AMHA.


We now have a sales incentive web site that reinforces the Peroni USA brand image, provides incentives and motivates our USA sales force.
Bright Rewards was a true collaborator in developing the online portion of our annual sales incentive. Their leadership and "end-user approach" helped the Peroni brand stay top-of-mind with our top beer distributors AND our field salespeople. Bright Rewards was very responsive, remained focussed on the plan objectives and exceeded my expectations. These guys are good!


If there is one existential truth about baseball players, it's that they're wildly inconsistent. Sometimes players will be league average or below, and then suddenly break out, for one year or for the rest of their career. Other times a strong player will have an off year or two before bouncing back (hopefully). These patterns in season-to-season output can be the result of a myriad of factors, including mechanical changes, aging curves, psychology, and luck, each of which teams employ small armies to study. While each player will be affected by each factor differently, the average player, over a large enough sample size, should experience similar trends in year-to-year production. My goal here is to try and measure those trends, and figure out the likelihood of a contextless player having a breakout, bounceback, or fallout season.
In this article, you're going to see a ton of scatter plots. For a multitude of statistics, I will compare players' output in a forth year with their stats from each of the previous three, and from averages of the previous three. I did this by writing a Python program to sift through Fangraphs leaderboards for players that qualified four years in a row, and then comparing their numbers. While this requirement gives my group a significant bias in talent level (average WAR 3.7), the year to year difference in talent with these players should be similar to that of the average player, especially considering that my data goes back to 2010 and covers a large sample of 200 hitters and 113 pitchers.
So, let's start by examining the overall trends. As predicted, between the three individual years, numbers from one year ago can best predict the output for this year. But, in almost all cases, stats from two or three years prior actually have about equal value. In the few exceptions, like batter WAR, the previous two years had similar correlations, with a significant drop-off after that. But, with every statistic, the averages were the most predictive of coming season's outcome. While the three-year averages were better than the two-year averages, they were also consistently only very slightly better. In fact, I'd say that for the everyday fan, taking a two-year average is your best bet with minimal effort. The ideal player predictor, then, would probably be a two year average with a slight weight on the most recent season, but when I tried to use machine learning to determine the weight, I got some really weird results. For example, on the three-year-average, it always gave the most recent seasons the same weights, and the stats from three years ago the biggest weights. Perhaps the machines are already smarter than us, and my computer is working in ways I cannot understand. But, a computer uprising notwithstanding, it's a safe bet to say that my machine learning algorithm was a failure. So, I can't provide you with exact weights, but as a general rule, my research shows that a two-year average is best for predicting whether or not a player will succeed in a coming season.
A few statistics had more irregular results than others. WAR, for example, had similar correlations between one year ago and two years ago. I would probably attribute this to the fact that it is composed of so many different statistics. If each of those stats has its own pattern of positive and negative regression, at least in part independent of the other components of WAR, then it would stand to reason that WAR as a whole is more variable year-to-year than each of those individual components. Therefore, last year's WAR makes a worse predictor of the coming year, and a more similar predictor to two years ago, at least in comparison to any of those components. Fangraphs' defense metric also had a somewhat strange outcome, where each year was a similar predictor, instead of last year being the best predictor, but that makes sense for defense in particular, because it is more ability-driven than hitting or pitching, which are more luck-driven. Earned Run Average also experienced strange correlation patterns. This could have a similar explanation to WAR, a run being composed of many different factors, but all in all I would attribute this strange pattern, along with overall low correlations, to ERA being a garbage statistic, even in my large sample size. However, among all of these different exceptions, it still held true that a two-year average is the best tool to use, by a meaningful margin over any single year. One stat that surprised me was HR/9 for pitchers. Sabermatricians have long heralded homers as one of the "three true outcomes," but it showed an both overall low correlation and a weird pattern, with the one-year and three-year P-Values being equal, and the two-year value being the lowest. Additionally, the averages only improved this P-Value by a little. Unlike the other exceptions, I can't easily explain away why this is happening. Maybe statisticians should consider relying less on home runs given up by pitchers to evaluate an arm. HR/9 turned out to be a notable exception, but for predicting most any other stat – power, contact, speed, control, or command – to gain an advantage, take a two-year average.
You don't u understand the data you're seeing here young man.


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De lijst van de hoogste gebouwen in Frankfurt am Main bevat informatie over de 25 hoogste gebouwen in Frankfurt.

Lijsten

Bestaande gebouwen

In aanbouw

Goedgekeurd

Voorgesteld

Zie ook

 Bankenviertel
 Lijst van hoogste gebouwen van Europa

Referenties
  Structurae
  SkyscraperPage

Frankfurt
Hoogste


New Delhi: India, with 15% of its population turning 40 over the next decade (age between 30-39 as of 2015), is expected to witness a high growth in healthcare. It is the size of this group of middle-aged people that drives growth in healthcare spending. At that age, healthcare spending starts to rise rapidly, only to slow when we get closer to 70 years old, according to researchers.
Healthcare spending tends to start to rise after 40. By that age, the body starts to show wear and tear and the prevalence of many health issues starts to rise when people turn 40. Diabetes is such an example, as illustrated for Canada. US data is very similar. It shows that the prevalence of diabetes rises from only 1.5% for the age group up to 44 years old to 12% for those aged 44-64. It rises further to 22% for those aged 65-74 and falls to 19% thereafter. "Thus, expect high growth in healthcare in younger populations, such as India," states a recent report by HSBC Global Research.
In emerging Asian markets (including India), healthcare spending is still low compared to the developed world – less than 5% of GDP vs 10% in Japan and 17% in the US. India's healthcare spending, at 4.7% of GDP (in 2014), is higher than in Indonesia and Malaysia, but there is still significant potential for further growth. All Asian markets have seen at least double-digit growth in healthcare spending as a percentage of GDP over the last two decades.
India is not only large, but also has quite a few people turning 40 in the next decade. These 40-something couples are enthusiastic consumers and are also becoming far more health conscious.
Domestic demand for healthcare is already on the increase. Hospitals are often not easily accessible for many rural Indians, increasing the attraction of homecare for chronic illness and post-operative care. According to Pacific Bridge Medical, a consultancy, an estimated 65% of hospital visits in India involve healthcare that can be done in the home. Still, hospitals are a more direct beneficiary of higher healthcare spending in India.
"We expect the focus (of private hospitals) to be on improving profitability after several years of expansion. For example, Apollo Hospitals is at the end of a three-year capex cycle and has added 1,725 beds in nine locations across India. The company plans to add another 480 beds in Mumbai and 65 beds in Indore in FY17," write Herald van der Linde and Devendra Joshi in the report, titled 'Making money from Asian demographics | The Flying Dutchman'.
Indian pharma companies are also exporters of products. The US is the main growth market, constituting over 45% of sales for some of the larger companies. "Our analyst believes a higher number of US FDA drug approvals in 2017 will benefit India's pharmaceuticals companies. A large domestic market offering stable growth also helps," the researchers write.
"Looking forward, I am confident that with better services and superior medical capability we will continue to play a very significant role in the fast evolving landscape for healthcare delivery in the country," added Singh.
Cygnus, comprising 10 hospitals across 3 states, offers free medicines and targeted need based discounting of pharmacy purchases, specialist consultations, and surgeries to the economically less advantaged directly combating the problem of healthcare costs as a route to poverty. In 2016-17 Cygnus enabled access to healthcare for the poorest by underwriting $1 million in cross-subsidized services, establishing itself as a brand committed to economic justice, according to Dr Bajaj.
Cygnus Hospitals has also forged cross-border alliances to bring 'best in class' training to India's medical community. For example, Cygnus partners with Global Health Alliance, UK to deliver specialised training at heavily subsidized rates such as ECG and echocardiography courses for health practitioners and paramedics, said Dr Bajaj.
Cygnus also plans to bring emergency healthcare to 50 small cities and towns of India, creating first time access to low cost super-specialty tertiary healthcare for 10 million Indians by 2030.


I can't resist dropping into a second hand book shops when I'm travelling or to the local one here in Blenheim. Naturally I head for the fishing and hunting section and recently in Blenheim I picked up for just $8 a copy of "Fishing the River of Time" by Tony Taylor. It tells of the author's experiences fishing with his grandson on Vancouver Island. In preparing himself at age 80, go take his eight year old grandson fishing.
Much the book is the lead-up to meeting his grandson. In the prelude, Tony Taylor reflects on fishing and what it is really all about. He recalls past fishing on the island with warmth and reflective wisdom. Trout fishing is not just about catching a trout but the total experience of going fishing.
I found much to agree with the author. For instance "A fisherman has to be an optimist: perhaps this is more important than catching fish." Or his criticism of the modern "high tech" gadgetry.
"If anyone asks me why I don't use more modern gear, I usually say it's too expensive for me. I don't tell them that most of the time it doesn't do what the advertiser says it will and if it does one becomes guilty of overkill. It's like using an elephant gun to shoot rabbits: it's showing off." " A hundred years ago anglers were obsessed with numbers but today its size. The truth is neither is important, but fishing is." And so on.
A good book like "Fishing the River of Time" is good for a dour damp drizzly winter's night. Its mention of Vancouver Island conjured up memories of Roderick Haig-Brown's books. Haig-Brown was so classy as a writer. If you haven't read them, seek out Roderick Haig-Brown's wonderful "A River Never Sleeps". His others are great too such as "Fisherman's Summer", "Fisherman's Winter" and "Fisherman's Fall." To get a copy or copies, search second hand bookshops or try googling and perhaps seek out sellers like Amazon?
Roderick Haig-Brown was an avid fly-fisher, pioneering conservationist, acclaimed author and magistrate. His writing – 25 books and well over 200 articles and speeches – have influenced fisheries biologists, ecologists and countless others interested in the evolving relationship between people and nature. Haig-Brown was born in England in 1908, and came to Canada in 1926 at the age of 19, and initially worked in logging on eastern Vancouver Island. He then worked for a period in the state of Washington, where he met his wife to be, Ann Elmore, who he married in 1934. They moved to Campbell River on Vancouver Island in 1936. The river property was ideal for the pursuits they enjoyed, particularly as Roderick was a devoted fly fisherman. He not only loved fishing, but was concerned about the welfare of the fish in the river and the ruffler itself, especially when Campbell River was experiencing a period of growth and new projects like a hydro dam were in the offing and were threatening the natural environment.
If you haven't worked it out yet, Haig-Brown is a favourite author of mine. Haig-Brown fought for the environment, for rivers and their fish. In 1965 in a barn-storming speech, he publicly targeted short-sighted politicians and government. He told the Canadian Authors Association at Victoria's Empress Hotel that he was appalled at "the shoddy, uncaring development of our natural resources, the chamber of commerce mentality which favors short-term material gain over all other consideration, the utter contempt for human values of every kind." By any measure, Roderick Haig-Brown was a strong and visionary voice against those who were ignorant of the ideals of conservation. As one tribute said, he listened to what the rivers and forests --ultimately what the salmon and other fish told him was to love and care for this finite planet: "Man must make himself small and humble to live within it rather than a ruthless giant to conquer it," he said.


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Q: What's the best way to integrate data from Kafka to HBase&Hive? I am new to this topic. I generate a lot of data in Kafka with time series data. And I need to send this data in a database either HBase or Hive. However,  I read that it is good practice to use Hive and Hbase together.
Kafka can connect to HBase and also to Hive with Kafka Confluent.
What is the best practice to send data from Kafka to HBase and then make a Hive HBase Integration or send data from Kafka to Hive and then make a integration with HBase?
Kafka -> Hbase -> Hive
or
Kafka -> Hive -> Hbase
And after that send this data to Spark.

redpajama_set_name: RedPajamaStackExchange


text:
Chehel Mani or Chehil Mani () may refer to:
 Chehel Mani, Anbarabad, Kerman Province
 Chehel Mani, Qaleh Ganj, Kerman Province
 Chehel Mani, Khuzestan
 Chehel Mani, Sistan and Baluchestan


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Thunderhead war eine deutsche Hard-Rock- und Metal-Band, die von 1988 bis etwa 2000 existierte. Als Fortsetzung wurde 2002 Donnerkopf gegründet. Heute ist ein Teil der Mitglieder als Nitrogods aktiv.

Bandgeschichte
Thunderhead wurde 1988 von Henny Wolter (Gitarre, ex-) und Ole Hempelmann (E-Bass) gegründet, die auf den US-amerikanischen Sänger Ted Bullet trafen. Bullet weilte in Deutschland, weil er ursprünglich den Sängerposten bei Victory antreten sollte, man jedoch erst in Deutschland feststellte, dass die Stimme nicht zur Band passte. Nachdem man zunächst mit Fritz Randow von Victory probte, wurde schließlich Alex Scotti Schlagzeuger der Ursprungsbesetzung. Mit einem ersten Demo bewarb man sich bei verschiedenen Plattenfirmen und erlangte schließlich einen Vertrag bei Intercord, wo im Februar 1989 das Debütalbum Behind the Eight-Ball erschien. Produzent war Will Reid Dick. Das Album verkaufte sich circa 30.000 Mal und die Band ging als Vorband für Uriah Heep und Victory auf Deutschland- und als Vorband von Motörhead auf England-Tournee.

1990 erschien das zweite Album Busted at the Border, das die hohen Erwartungen an den Nachfolger von Behind the Eight-Ball nicht erfüllen konnte und lediglich 10.000 Mal verkauft wurde. Die Band machte dafür zum einen den schlechten Vertrieb, aber auch die kraftlose Produktion von Tony Platt verantwortlich. Die darauffolgende Headliner-Tour wurde ebenfalls ein Misserfolg.

1991, nach dem Desaster mit dem letzten Album, beschlossen Thunderhead die Produktion alleine zu übernehmen. So wurde Alex Scotti letztlich Produzent des Albums Crime Pays. Das Album erschien über Music for Nations. Es folgte eine Deutschland-Tour mit Gamma Ray sowie eine sehr erfolgreiche Japan-Tournee. Zurück in Deutschland spielte man eine weitere Headliner-Tour mit Co-Headliner Crossroads. Das folgende Album Killing with Style erschien schließlich 1993, nach längerem Streit mit der ehemaligen Plattenfirma Intercord, erstmals über Gun Records und wurde zum erfolgreichsten Album der Bandgeschichte. Zur Singleauskopplung Young & Useless wurde ein Video gedreht. Die anschließende Tournee fand mit Hauptband Saxon statt. Es folgte außerdem eine Japan-Tournee mit Rage. 1994 erschien außerdem das Live-Album Classic Killers – Live.

Für 1995 waren zwei Alben geplant, ein Album mit schnelleren Stücken und ein Balladenalbum. Zunächst erschien Were You Told The Truth About Hell? als härteres Album, nebenbei wurde an The Ballads '88 – '95, das letztlich jedoch eine Kompilation mit alten Liedern, zwei neu eingespielten sowie zwei neuen Songs wurde. Mit Behind Blue Eyes wurde auch eine Coverversion von The Who eingespielt. Nach einem Best-of-Album 1999 ließ Gun Records die Band jedoch wegen zu geringem Absatz fallen. Die Band versuchte zwar noch einen Folgedeal auszuhandeln, doch zerbrach in der Zwischenzeit auch das Bandgefüge. Über Steamhammer Records erschien das letzte Werk der Band, Ugly Side, danach verließ Ted Bullet die Band, um sich auf Sonofabitch zu konzentrieren, die jedoch ebenfalls im gleichen Jahr ihre Tätigkeiten einstellten.

Später versuchte Bullet ein Comeback mit neuen Mitgliedern, das jedoch letztlich ebenfalls scheiterte und ohne Veröffentlichung blieb. Seine neuen Mitstreiter unterstützten später das Soloprojekt Wayne des Ex-Metal-Church-Sängers David Wayne.

2003 versuchten Ole Hempelmann und Henny Wolter zusammen mit Klaus Sperling mit der Band Donnerkopf an die Erfolge der 1990er Jahre anzuknüpfen und setzten dabei auf deutsche Texte. Es erschien ein Album über Teuton. Henny Wolter ist seit 2012 zusammen mit Klaus Sperling und "Oimel" Larcher mit den Nitrogods aktiv.

Musikstil
Thunderhead spielen simplen und eingängigen traditionellen Metal an der Grenze zum Hard Rock, ohne viele Schnörkel oder viele Soloeinlagen. Der Musikstil entwickelte sich später hin zum Power Metal. Ted Bullet wird als energetischer Sänger beschrieben, dessen tiefe, raue Stimme und direkte Texte perfekt zur Musik passten.

Diskografie

Alben
 1989: Behind the Eight-Ball (Intercord)
 1990: Busted at the Border (Intercord)
 1991: Crime Pays (Music for Nations)
 1993: Killing with Style (Gun Records)
 1995: Were You Told the Truth About Hell? (Gun Records)
 1999: Ugly Side (Steamhammer Records)

Singles und EPs
 1989: Take It to the Highway (Intercord)
 1989: The Fire's Burning (Intercord)
 1990: 25 or 6 to 4 (Intercord)
 1993: Young and Useless (Gun Records)
 1993: Movin' On (EP, Gun Records)
 1995: Snap (Gun Records)

Kompilationen und Live-Alben
 1994: Classic Killers Live (Gun Records)
 1995: The Ballads '88 – '95 (Gun Records)
 1997: The Best of Thunderhead (Victor Entertainment, Japan-Veröffentlichung)
 1999: The Whole Decade (Gun Records)

Sonstige Veröffentlichungen
 1991: Metal Beat Messiahs (Split-CD mit Zodiac Mindwarp, Night Shade und Hades, Promo-CD vom Rock Hard)

Einzelnachweise

Deutsche Band
Metal-Band