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I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences …

Quadratic (or bilinear) forms appear naturally throughout mathematics, for instance via inner product structures, or via dualisation of a linear transformation, or via Taylor expansion around the line …

I did a preliminary feasibility analysis of our methods and it appears possible that one may be able to tighten our $n^\epsilon$ loss to something more like $\exp( \sqrt{n} )$ in the Gaussian case, bu …

Shorn of probabilistic language, this inequality follows from the assertion that $|x+y|-|x-y|$ is a positive semi-definite kernel, and is therefore the sum (or integral) of squares. Your Fourier-anal …

I adapt an argument from this blog post of mine, exploiting the $\ell^2$ boundedness of the discrete Hilbert transform (i.e. Hilbert's inequality), to obtain an exponential upper bound. I don't see a …

The similarity between the entropy power inequality and the Brunn-Minkowski inequality is not directly related to convexity - after all, Brunn-Minkowski can be generalised to bounded open sets that ar …

The theory of classical independence and classical convolution can be generalised to noncommutative settings in several ways. The most famous one is that of free independence and free convolution (int …

Quite a lot of questions here! It is perhaps worth making a distinction between scalar classical probability theory - the study of scalar classical random variables - and more general classical proba …

Firstly, the equation you attribute to Silverstein (and is sometimes known as the "self-consistent equation" for the Stieltjes transform) is not exact, but only asymptotically valid in the limit $n \t …

After chasing down references relating to the paper of Shelah mentioned by Will Brian, I now have a satisfactory answer to the question. It all hinges on whether there is a splitting of the quotient …

The answer is no. This is a good illustration of a reasoning principle identified explicitly in Gowers, W. T., The two cultures of mathematics, Arnold, V. (ed.) et al., Mathematics: Frontiers and pers …

One heuristic is to replace the $n^{th}$ roots of unity by $n$ iid elements $\zeta_1,\dots,\zeta_n$ of the unit circle, drawn uniformly at random. For any sum $\zeta_{i_1} + \dots + \zeta_{i_k}$ of $ …

The Lovasz-Szegedy theory of graphons is likely to be relevant. Every measurable symmetric function $p: [0,1] \times [0,1] \to [0,1]$ (otherwise known as a graphon) determines a random graph model, i …

This appears to be the case, but I was forced to rely on a somewhat complicated inequality on two real variables that looks quite plausible numerically, though I do not have a 100% rigorous proof of i …

Bounded density will suffice, I think. Basically what one needs is for the Fourier transforms (aka characteristic functions) of the $X_1 + \ldots + X_n / \sqrt{n}$ to converge pointwise to the Fourie …