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In Ricci curvature of Markov chains on metric spaces Yann Ollivier, defines a coarse Ricci curvature for a Markov chain with transition kernels $\{m_x\}$ defined on a metric space $(X,d)$ as follows: …

$\newcommand\Ac{\mathcal A}$ $\newcommand\BL{\operatorname{BL}}$ $\newcommand\reals{\mathbb R}$ $\newcommand\eps{\varepsilon}$ $\newcommand\pr{\mathbb P}$ $\newcommand\ex{\mathbb E}$ $\newcommand\give …

Let $(\Omega,\mathcal A, P)$ be a probability space and let $\Sigma(\mathcal A) \subset 2^{\mathcal A}$ be the collection of all sub-$\sigma$-fields of $\mathcal A$. Then, $\Sigma(\mathcal A)$ is clos …

Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as $$ d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2} $$ wh …

Let $\{\varepsilon_i\}_{i=1}^n$ be a sequence of independent Radecmacher (i.e., symmetric Bernoulli) variables, and let $\phi_i :\mathbb R \to \mathbb R$ be contraction (i.e., 1-Lipschitz) mappings th …

Filling in the details for Anthony's argument: Assume that $\mathbb E |X|^3 \le c$ for numerical constant $c > 0$. Let $X^+ = X \mathbf 1_{X > 0}$ and $X^- = (-X) \mathbf 1_{X < 0}$. Then, $X = X^+ …

Assume that $X_n$ is a sequence of a zero-mean and unit variance random variables (and maybe having density w.r.t. to Lebesgue). Can we conclude that $ P(X_n \in [0,R_n]) $ is bounded away from zero e …

Let $x \sim N(0,I_n)$. For any independent rank-1 projection $A$, conditioned on $A$, we have $$x^T A x \sim \chi^2_1.$$ So unconditionally, $x^T A x = O(1)$ with high probability. Now, let $A = \fr …

Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality \begin{align*} |f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n. \end{align*} For $n \ge 2$, can we f …

Consider an independent collection of random variables $W_i, i=1,\dots,n.$ and let $Z \sim N(0,1)$. Roughly speaking, we know that $W_i$ are close in distribution to $Z$, say each is itself a sum of $ …

Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If $|h(w_1,w_ …

I believe you can do something like this: Let $X$ be zero mean. Then the covariance matrix is $ \text{cov}(X) = \mathbb{E}[XX^T]$. Hence, by the same argument: \begin{align*} \text{cov}(X) = \text{ …

Let $(p_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and …

Let us consider convex polytopes with $K$ extreme points in $\mathbb{R}^d$. Let $\mathbf{P}$ be such polytope, and let $\text{ext}(\mathbf{P})$ denote its set of extreme points, so that $\mathbf{P} = …

Let $X_t$ be the number of points in $[0,t]$. Then, $X_t$ is a renewal process. Let $m(t) = E[X_t]$. Then renewal theorem says $$ m(t+h) -m(t) \stackrel{t \to \infty}{\longrightarrow} \frac{h}{\mu} …