## Tagged Questions

3answers
193 views

### how to proof this Stirling related equation

here is what I need to proof, have no idea were to start. I know there is some connection with the Stirling theorem.  \sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right …
0answers
109 views

### Bounding a sum of binomial coefficients in terms of ‘the next one’

I need to bound a sum of a portion of binomial coefficients in terms of "the next one", and understand what is the best which can be said in this sense. Given a real number \$t \ge …
1answer
221 views

### A spectral inequality for positive-definite matrices

Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues $$\lambda_1 \leq \cdots \leq \lambda_n ,$$ is there a sharp upper bound for the product \$ …
1answer
180 views

### A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and  K^* := \{y \in \mathbb{ …
1answer
121 views

### Inequality of Partial Taylor Series

Hi, For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: \sum_{k …
0answers
238 views

### Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so afte …
1answer
385 views

### The fraction of the sphere a fixed distance from a subspace

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance a …
0answers
85 views

### Tight Upper/Lower bound for Incomplete Gamma function

Does anyone know of any tight upper/lower bound for incomplete Gamma functions? i.e either of the following functions:  \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t …
0answers
77 views

### Distribution of convex combination of i.i.d Gamma random variables

Hi, I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider $x_{i}$ be $Gamma(\theta,1)$, then would we have the …
1answer
477 views

### What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities: If $k < n/p$ then $u\in L^q(U)$ for \$ …
0answers
21 views

### Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors: \begin{eqnarray*} \bar{a} &a …
4answers
305 views

### Understanding Gibbs’s inequality

Short version Gibbs's inequality is a simple inequality for real numbers, usually understood information-theoretically. In the jargon, it states that for two probability measures …
2answers
175 views

### Spherical Bessel functions

I wish to show that $|j_n(x)| < \frac{1}{\sqrt{x}}$ for $n=0,1,2,\ldots$ and $x>0$, where $j_n$ is the spherical Bessel function of the first kind. Experimenting with Matlab …
1answer
107 views

### Chernoff-Hoeffding bound for complex values

Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value $\mu$ and satisfying $|X_i| \le b$. Le …
1answer
198 views

### An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function: \begin{equation} g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k. \end{equation} Prove that $\exists C>0$ and …

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