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The Kullback-Leibler divergence is a special case of Rényi divergence. In your notation, for $\alpha > 0$, the Rényi divergence of order $\alpha$ is defined by $$ D_\alpha(p_0,p_1) = \frac{1}{\alpha …

One often denotes $\psi_\alpha (x) = e^{x^\alpha} - 1$ for $\alpha \ge 1$, and the corresponding Orlicz space is $L_{\psi_\alpha}$. An estimate on the $L_{\psi_\alpha}$ norm is also often referred to …

In the spirit of this answer to a different question, I'll offer a contrarian answer. How to understand probability theory from a structuralist perspective: Don't. To put it less provocatively, what …

This old question and my old answer continue to get occasional attention, and I believe it's time for a new answer. Is there an introduction to probability theory from a structuralist/categorical per …

I don't know of any systematic study of such symmetries in any great generality. On the other hand, as in most (if not all) fields of mathematics, probability theory happily exploits symmetries to he …

Your question is part of what's called Littlewood-Offord theory, which has seen a lot of progress lately in work of Tao and Vu and of Rudelson and Vershynin. Take a look at Section 1.2 and especially …

As you suggest, the fact that $e^x$ is increasing is a useful property here. In particular, that lets you in some cases to apply Markov's/Chebyshev's inequality to the random variable $e^{tX}$ in ord …

The easiest condition would be a bound on $\sup_i \mathbb{E} X_i^6$, which would allow you to apply the Berry–Esseen theorem. More generally, if for some $0<\varepsilon < 2$ you have a uniform bound …

A little too long for a comment, but I don't have time right now to turn this observation into a proper answer: Another way to generate a uniform random point in the simplex is to let $b_1, \ldots, b …

Let's try again. The theorem of Otto and Villani implies that every distribution $\nu$ which is log-concave in the sense you define satisfies a transportation-cost inequality. There are many distrib …

Carlo's answer addresses the first question. To address the second one: the "disk law" (better known as the "circular law") does not tell you the distribution of singular values. However, the Marchen …

I just came across this old question, while searching for whether it has been proved yet that spatial determinantal point processes are negatively associated. It turns out it has been proved, in this …

I don't think there's an exact expression, but the Bai–Yin result does give the right prediction. It's a little easier to state nice-looking results for a $p \times n$ matrix $X$ with independent sta …

It depends on exactly what you mean by "of the form ($*$)". As Davide points out (and as you certainly know if you've been reading Boucheron, Lugosi, and Massart), for centered subgaussian random var …

You don't find much about time-inhomogeneous Markov chains because it's extremely difficult to prove anything about them without strong additional assumptions, and it's not clear what additional assum …