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Let $X \in \mathbb{R}^{n \times d}$ be a random matrix with iid standard Gaussian entries. Let $e_1$ denote the first canonical basis vector in $\mathbb{R}^d$. Define $$ P_d(\lambda) = (1-\lambda) e_1 …

Suppose that $X$ is distributed uniformly in the scaled $n$-sphere $\sqrt{n} \mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X_1, \dots, X_k)$, the first $k < n$ coordina …

Let $A_n$ be a random $n \times n$ matrix with entries in $\{-1, +1\}$. As usual, "random" here means with respect to the uniform measure over such matrices. The strong version of the singularity conj …

On page 5 of a recent manuscript by Lugosi-Mendelson, a claim equivalent to the following statement is made: Suppose $Z$ is a centered, $\mathbf{R}^d$-valued random variable with $\mathbf{E} e^{\lamb …

Let $\mathcal{M}_1$ denote the set of probability measures on the unit ball in $\mathbb{R}^d$ (which comes with its Borel $\sigma$-field). Denote by $\sigma$ the uniform measure on the orthogonal grou …

Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices. Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have it …

Let $X$ be a binomial random variable with parameters $n,p$. Define the function $f(n, p, t) = E\frac{1}{1 + t X}, $ where $t > 0$. Question: Can we find an elementary function $F(n, p, t)$ such that …

Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$. Consider the ratio (for $k \geq n$) $$ R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k …

Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix $$ A = I_n - uu^T. $$ Question: What techniques are available to provide (reasonably g …

Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries. In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ …

The DKW inequality says that if $F_n$ is the empirical CDF corresponding to real-valued random variables $X_1, \dots, X_n$ distributed identically and independently from a distribution with CDF $F$, t …

Let $\{U_{k, n}\}_{k=1}^n$, denote the order statistics of a sample of $n$ iid uniform $[0, 1]$ variates. Note that, marginally $U_{k, n}$ is distributed $\mathrm{Beta}(k, n+1 -k)$. Therefore, let us …

Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm. Let $x \in \mathbb{R}^n$ and define $$ F(x) = \mathbb{E}[\|x + g\| - \|g\|]. $$ I am wondering if it is possible to …

Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$. Define $$ F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big] $$ If $\|x\|_\infty …

In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable. Now suppose that $x$ is a (say, centered) …