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Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding $$ |\mathbf{E}_{X, Y\sim P \otime …

In the paper "Spin glasses and Stein's method" (https://arxiv.org/pdf/0706.3500.pdf), Sourav Chatterjee established Stein's equation for mixtures of two Gaussian densities in $\mathbb{R}$, which takes …

Suppose we have two Gibbs measures with densities $$ p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)). $$ Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, …

Suppose that we have a Markov process $\{Z_t\}_{t=0}^\infty$, where $Z_t \geq 0$ for any $t$. Assume that, conditioning on $Z_t = z_t$, we have $ \mathbb{E}\{Z_{t+1}|Z_t = z_t\} \leq \kappa z_t^2 $. …

We consider the two distributions $$ p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I), $$ where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and varianc …

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector …

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a Gaussia …

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a s …

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where $$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j …

Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which $ …

Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/Hierarchical_hi …

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in …

Let $U, U'\in R^{d\times k} (d>k)$ be two independent uniformly distributed random orthonormal matrices. In specific, let $S$ be the set of all $d\times k$ orthonormal matrices. Here 'uniform' is defi …

The high-level question is: can we generate any random graph with size $d$ using a Markov chain? For example, let $X^{(0)} = (1,0,\ldots,0) \in R^d$ be the initial state, and $X^{(t+1)} = f^{(t)}(X^ …

Let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., one that has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the …