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You can directly use Chernoff's inequality, to get $LHS \le e^{-\Lambda^*(\epsilon)}$, where $\Lambda^*(\epsilon) := (\epsilon-1-\log\epsilon)/2$ is the Fenchel-Legendre transform of the log-MGF $\Gam …

It looks like all the previous answers (including mine) are quite off by a huge margin, if the vector $a=(a_1,\ldots,a_n)$ is somewhat dense in the sense that $\|a\|_1/\|a\|_\infty$ is substantially l …

There are a few references worth mentioning: https://hal.archives-ouvertes.fr/hal-00915365/document https://www.lpsm.paris/pageperso/bolley/bgv.pdf https://arxiv.org/pdf/1804.10556.pdf Example of …

Although great answers have already been provided, the one provided below is perhaps be the shortest possible answer to the question. Let $x_1,\ldots,x_k \in \mathbb R$ such that $r_k^2:=\sum_{i=1}^k …

Let $(X,\mu)$ be a probability measure space and $A$ be a measurable subset of $X$ such that $0 \le \mu(A) < p < 1$. Question When is it true that there exists a measurable $B \subseteq X$ such that …

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 $\mu$ be a probability measure on measurable space $X=\mathbb R^n$ (euclidean), and let $F$ be a family of $\mu$-measurable functions $X \mapsto \mathbb R$ which are uniformly bounded, i.e $b:=\su …

The answer to your question appears to be contained in a 1928 Theorem of Polya, on simple random works on a lattice $\mathbb Z^d$. As a soft-reference, see page 15 of this short paper, using only very …

Below, I provide a "high-probability" non-asymptotic bound (see (+) below) based on non-linear Berry-Esseen theory developed by Iosif Pinelis. I'd be grateful if someone would kindly check that I didn …

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 …

As pointed out by a user (Nate Eldgredge) in the comments under the question, $$ P\left(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^n a_i^2Z_i^2 \ge \epsilon\right) = P\left(\max_{1 \le i \le n}Z_i^2 \ge …

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 $(X, \mathcal F, P)$ be a probability space. Question What kind of condition is this: there exists a sequence $(a_n)_n \subseteq X$ such that $\forall$ measurable $A \subseteq X$, $P(A) > …

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 …

You can prove it via standard computations. See (2.1), (2.7), and (2.8) of On the maximum of the generalized Brownian bridge.