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Let $\nu \propto e^{-f}$ be a probability density on $\mathbb{R}^d$ with full support. We say $\nu$ satisfies the log-Sobolev inequality (LSI) with constant $\alpha$ if for every smooth function $g:\m …

Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random wa …

Let $X_1, \ldots, X_n$ be independent and identically distributed random variables. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a bounded difference function, i.e., for any $x,y \in \mathbb{R}^n$ that diff …

For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form: $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left( …

For $i= 1, \ldots, n$, let $A_i \in \mathbb{R}^{d \times d}$ be random i.i.d. matrices with $E [A_i] =0$. Can we relate (upper bound) $E[\|\sum_{i=1}^n A_i \|_F^4]$ to $E[\|A_i\|^4_F]$ ?

Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for $$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, …

Stein's Lemma in its standard form states that $X \sim N(0,1) \Leftrightarrow E[f'(X) - X f(X)] =0 $ for all bounded one-time differentiable functions $f$ (I am ignoring the exact conditions on $f$ fo …

Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with inne …

Let $Y=X^\top W$ , with $X, W \in \mathbb{R}^{d \times d}$ are random matrices with standard normal entries. Let $\lambda_j$ be the $j^{th}$ singular value of $Y$. Is there a way to bound the tail pro …