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A possible relevant post What kind of random matrices have rapidly decaying singular values?. In that post I discussed the distribution of maximal eigenvalue of a random matrix based on the result [Jo …

This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? and current heated research front of high dimensional sta …

The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\si …

Yes, but with the price of additional assumptions on variances $\sigma^2$ of samples $X_t$([1]'s proof can be extended to non-identical variance $\sigma ^2_i$). For example, [1] Theorem 2 strengthens …

A great example that I have in mind is the concentration phenomena in high dimensions. Consider the simplest multivariate normal distribution $X\sim N_d(0_d,I_d)$, we can compute its $L^2$ norm $\sum_ …

I do not think the conclusion will hold for a general exchangeable sequence. In a more general case, you have to assume that U-statistics itself are not degenerated ($w_i\neq w_j$ for $i\neq j$) and t …

This argument based on modification of [2]. (1) Let A be the event that $\frac{1}{n}\sum_{i=1}^{n}X_{i}\geq\frac{1}{2}$ , let $X_{i}=0$ mean heads of coin and $X_{i}=1$ mean tails. $Pr(A)\leq Pr\lef …

To take a different view of the question, what you are asking is whether the sample paths of this Wiener process $W_0$ can be "more smooth" in terms of Hölder continuity. The answer is NO. To make @M …

I think a closest inequality in controlling the deviance between two processes is McDiarmid’s Inequality, and it has an extension which also applies to subgaussian random variables Concentration in un …

The expectation can be computed in closed form, and I think that without further assumptions on entries of the matrix $Y$, the Jensen bound is sharp according to following calculation: $\begin{align}\ …

Iosif Pinelis provided a very nice answer, however, I would like to provide a more comprehensive answer to this question. I think the title is a bit misleading because we do not actually need the orde …