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It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a stro …

Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with. So, my understanding is that category theory and related fields of higher mathematics …

Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid $$\operatorname{KL}(p_\theta\p …

Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies Talagrand …

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by …

Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = …

Disclaimer. Question moved from SE. Setup Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$. Question What is a good upper-bound for $\mathbb E[|X-np|^r]$ ? Solution for small $r$ If $r=2$, the …

In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that $$ d(A, B) \le W(\mu|_A, \mu|_B), $$ where $d(A, B):= \i …

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally distribute …

Let $M > 0$, $k$ be a positive integer, and $\mathcal V:=[-M,M]^k$. Finally, let $p \in \Delta_k$, (where $\Delta_k$ is the $(k-1)$-dimensional probability simplex) and let $\hat{p}_n$ be an empirical …

Let $A$ be a nonempty measurable subset of $\mathbb R$, with Lebesgue measure $|A|=1$, and let $c>0$. Define the scalar $I(A)$ by $$ I(A) := \int_{-c}^c |A \cap (x + A)|\, dx, $$ where $x+A := \{x + a …

Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define $$ \alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(x)\|^k}\ri …

Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ upt …

Let $X$ be an $n \times d$ random matrix with iid entries from $N(0, 1/d)$. Let $S:=X^\top X/n$, a $d \times d$ Wishart matrix and let $T = e^{S} := \sum_{k=0}^\infty \dfrac{S^k}{k!}$ be its exponent …

In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the solutio …