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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 …

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 $W = X^TX$ 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 $D$ …

Let $\{x_i\}_{i\geq 1}$ be iid random elements of the sequence space $\ell^2(\mathbb{N})$; assume that $\|x_i\|_2 \leq 1$ almost surely. Let $\Sigma = \mathbb{E}[x_1 \otimes x_1]$. Define $$ \Sigma_n …

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 $X$ 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 $z_1, \dots, z_n$ 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 …

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 …

Consider the empirical and population CDF, $$ F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad F(t) = \mathbb{E} [F_n(t)], $$ where above $X_1, \dots, X_n$ are iid, real-value …

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 …

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 $\mathcal{S}_+^d$ 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 $\{\phi_k\}_{k=1}^K$ be a family of functions mapping from an interval $[a, b]$ to $[-1, 1]$. That is, $\phi_k \colon[ a,b] \to [-1, 1]$ are nondecreasing maps on some finite interval $[a, b] \sub …

Say $x$ is a random vector in $\mathbb{R}^n$. Then, given a (deterministic) symmetric real positive definite matrix $A$, if we want to calculate the expectation of the quadratic form, we can use the i …