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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 …

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 …

There's also a softer argument based on properties of the heat kernel, which applies in higher dimensions as well, in Lemma 4.9 of this paper of Klartag. It shows that in $n$ dimensions, the total va …

You can do something with Talagrand's inequality for at least some normalizations. The simplest case would be if the $X_i$ are mean 0 and bounded, say $|X_i| \le 1$ almost surely, and you're looking …

What counts as "best"? The smallest tail bound is of course $$ (2\pi)^{-n/2} \int_{\{(x_1,\dotsc,x_n) : \sum x_i^4 > (1+t)3n\} } e^{-\sum x_i^2 / 2} dx_1 \dotsb dx_n. $$ Presumably you want somethin …

The distributions you're looking for are stable distributions. Basically, the only such norms you can take are $\ell^p$ norms for $1 \le p \le 2$. If you don't need an honest norm, you can also take …

The usual approach is to generate $n+1$ i.i.d. mean zero Gaussian random variables $X_1, \dotsc, X_{n+1}$ to get a random point $X$ in $(n+1)$-space with rotationally invariant distribution and normal …

Without further assumptions you can't do better than the union bound (which should be $n e^{-\epsilon^2}$ as you've written things). If $X_i$ are identically distributed and the events $(|X_i| > \eps …

For large $N$ asymptotics, you want to look into extreme value theory. In particular, take a look at this book.