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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 …

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 subcollection $\mathcal S$ of $k$-element subsets of $[n]$, consider the random variable $Z_{\mathcal S} := \sup_{A \in \mathcal S}|X_A|$, where $X_A:=\sum_{i \in A}X_i$, and the $X_i$'s are iid …

Consider the "loss function" $\ell_t:\mathbb R^2 \to \{0,1\}$ defined by $\phi_t(y,y') := 1_{yy' \le t}$, and let consider the function class on $X \times \{\pm 1\}$ given by $$ S_t(F):= \ell_t \circ …

Disclaimer: This is just a detailed version of Aryeh's answer. So, let $\mathcal C_\epsilon$ be $\epsilon/2$-cover for $\Theta$ of minimal cardinality $N(\Theta;\epsilon/2)$. Let $C \in \mathcal C_\e …

Solution with added restriction that $h$ is Lipschitz continuous Below, I do some computations which seem to suggest the result is true under some additional smoothness constraints on $h$. I'm not 10 …

Below, I provide an answer inspired by the comments of user Terry Tao. Let $n/N =: \lambda \in (0, 1)$ be the aspect ratio of $A$. We will prove the following. Claim. For every $C>0$, there exists $ …

I managed to piece together a solution to my problem by reading the first page of this paper http://www-users.math.umn.edu/~bobko001/papers/2010_JMS-165_Conc.on.the.cube.pdf. I'll only handle the euc …

Claim (Nonasymptotic result under smoothenss condition). Suppose $g$ is $\mathcal C^5$ at $0$ and that $d'$ and $d$ are sufficiently large with $c_1 \le n'/d \le c_2$ for some absolute constants $0 < …

Here we bound the entire spectrum of $\Sigma'$, from below and above. This post is inspired by a comment of user @BrendanMcKay. Claim. $\lambda_\max(\Sigma') = O(m/n)$ and $\lambda_\min(\Sigma') = \O …

For any $c \ge 0$, define $\theta(s,c)$ by \begin{eqnarray} \theta(X,s,c) := \inf_{\delta \in \mathrm{CRE}(s,c)}\dfrac{\|X\delta\|_2}{\sqrt{n}\|\delta\|_2}, \end{eqnarray} where $\mathrm{CRE}(s,c) := …

It turns out that Neyman-Pearson theory helps get a nontrivial inequality. Notations. For a p.s.d matrix $M$ of size $p$, consider the inner product on $\mathbb R^p$ defined by $\langle x,z \rangle_M …

It turns our the problem has a simple answer, once the easy case has been solved (see the OP). Indeed, we write $\overline{f} = f + \zeta_0(f)$, so that $\zeta_0(f) = 0$. Now, one has $$ T = \overlin …

WLOG, let $\lambda = 1$ (rescale your problem appropriately, if necessary). Then, it is well-known consequence of Bernstein's inequality (e.g see theorem 3.1.1 of "High-dimensional Probability" book b …

It turns out it is possible to prove a (very very) slightly weaker lower-bound, namely $\|v\| = \Omega\left(\left(\dfrac{d}{k}\right)^{1/2}n^{1/2-o(1)}\right) \text{ w.p }1-o(1). $ Moreover, this bo …