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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 $\{x_i\}_{i\geq 1}$ 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 …

Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $1$-Lipschitz function. Define the uniform norm $\|f\|_\infty = \sup_{x} |f(x)|$. Given $\{U_j\}_{j=1}^\infty$ independent and identically distribute …

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 $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson …