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Let $X$ be a standard Wishart matrix, i.e., $$ X = \sum_{j=1}^n g_j \otimes g_j \quad \mbox{where} \quad g_j \sim N(0, I_d). $$ Above, $g_j$ are independent samples from the standard multivariate Gau …

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 $\mathcal{S}_+^d$ 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 $\{x_i\}_{i=1}^n \subset \mathbb{R}^d$ be iid random vectors drawn from probability measure $P$. Define the random $d \times d$ real positive semidefinite matrix, $$ S_n = \frac{1}{n} \sum_{i=1}^n …

Let $d < n$, and let $G_n(d)$ denote the space of all $d$-dimensional subspaces of $\mathbb{R}^n$. Let $a = (a_1,\dots, a_n)$ denote a positive sequence, and define $U(a) = \{u \in \mathbb{R}^n: \sum_ …

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 …

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 …

Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function $$ f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big], $$ over the class of real sy …

Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$). Consider the map $$ f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle, $$ define …

Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, …