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Let $A$ be a fixed $n$ by $n$ real symmetric positive definite matrix with eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n > 0$, and let $f(A):=\sum_{i=1}^n\log\lambda_i$, and let $X$ be …

Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X …

Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$. Question Given $\epsilon > 0$ (may be assumed to be very small), what is a r …

For a Borel subset $B$ of a metric space $X = (X,d)$ and $\epsilon>0$, recall the defintion of the $\epsilon$-blowup of $B$, namely $B^\epsilon = \{x \in X | d(x,B) \le \epsilon\}$. Let $\mu$ be a pro …

Let $\mathcal X=(\mathcal X, d)$ be a Polish space (i.e complete metric space), and let $\Omega$ be a non-empty subset. Consider the binary cost function $c_\Omega$ on $\mathcal X^2$ defined by $c_\Om …

A random variable $X$ is said to be sub-gaussian with mean $\mu$ and pseudo-variance $\sigma^2$ iff $$\mathbb \log(E[\exp(t(X-\mu))]) \le \frac{t^2}{2\sigma^2},\;\forall t \in \mathbb R. $$ It's a st …

Let $P=(p_1,\ldots,p_k) \in \Delta_k$ be distribution supported on set of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ based on an iid sample of size $n$. Question What's a good non-a …

Let $\Theta$ be a subset of a metric space. Suppose $(X_\theta)_{\theta \in \Theta}$ is a random process on $\Theta$ which is $L$-Lipschitz and with the property that there exists constants $A, B>0$ s …

Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R …

Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies Talagrand …

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ …

Let $Z$ be a random variable with finite moment-generating function $M_Z(\theta):=E[e^{\frac{1}{\theta}Z}]<\infty$ for all $\theta > 0$, and for $\delta \in (0,1]$, define $C_Z^\delta := \inf_{\theta> …

Let $x \in [-1/2,1/2]$ and $X_1,\ldots,X_n$ be drawn iid from the uniform distribution on $[-1,1]$. Question. Given $\varepsilon \ge 0$ an integer $k \in [1,n]$, what is a good lower-bound on the pro …

Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$. Q …

Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges w …