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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 …

Here is my solution without the reduction trick to $1$D gaussian. Let $U := X/\|X\|$. Since $U$ is uniformly distributed on the unit $n$-sphere, it follows that the random variable $U^Tz$ has the sam …

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 …

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 …

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 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 …

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 $ …

This is what I've come up with. It's too long to be a comment, so I decided to post it as an answer. So, it was proven in LeCam, L. M. (1969). Théorie Asymptotique de la Décision Statistique, p35 tha …

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 …

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. Rescale things so that $\mathbb E [c_{11}^2] = 1$. In the limit when $n,k \to \infty$ such that $k/n=\lambda \in(0,\infty)$, the spectral density of $C$ converges to $MP(\lambda)$. Proof. Fol …

It turns out that one can get a stronger result than demanded in the question: compute $\Delta(a,b)$ for any $a,b \in S_{d-1}$, perpendicular or not. Indeed, Claim. If $f(\rho)=a_0 + a_1 \rho + a_2 \ …

It turns out that the optimal value of the problem can be computed arbitrarily well using basic probability arguments. Of course, this post doesn't answer my question, since the only motivation of the …

I provide a complete solution for the case where $A$ is the intersection of $N = \mathcal O(\mathrm{poly}(n))$ half-spaces $H_i := \{x \in \mathbb R^n \mid x^\top w_i \le b_i\}$, where each $w_i$ is a …

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 …