Search Results

One can get a bound which is within a constant of the optimal bound using the following Paley-Zygmund type inequality Let $X$ be a real random variable with mean zero and finite fourth moment, th …

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 …

To complement my other answer, I will show Proposition 1 Let $\xi_k$ be a finite number of iid Bernoulli random variables of expectation $p > 1/2$, and let $a_k > 0$ be real numbers. Then ${\bf …

As observed in a (now deleted) previous comment, the exponent of $\|\alpha\|_2$ should be $2k$ instead of $2$ for homogeneity reasons. If the $\varepsilon_i$ are symmetric, then this can be proven by …

This is a non-centered iid random matrix whose entries have mean one and variance one (and decay exponentially at infinity), and as such, is subject to the circular law with one outlier. Thus, there …

Hmm, you're asking for concentration for heat kernels. Over long periods of time, these kernels are dominated by the low-energy eigenfunctions, so basically one needs to construct domains which have …

Let's normalise the variance of the entries to be $1$. Then GUE asymptotically obeys the semicircular law, i.e., the eigenvalues (which equal the singular values, as GUE is Hermitian), after dividing …

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 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 …

There is certainly no asymptotic independence between $\det M_{11}, \det M_{22}$. From the base times height formula for parallelepipeds we see that \begin{align*} \frac{|\det M_{12}|}{|\det M_{22}|} …

If one replaces the real line with the Walsh ring $F_2[t](\frac{1}{t})$ (or equivalently, replaces the Fourier transform by the Fourier-Walsh transform), then Littlewood-Paley projections become preci …

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 …

This is studied in Gérard Ben Arous and Paul Bourgade, Extreme gaps between eigenvalues of random matrices, Ann. Probab. 41 (2013), no. 4, 2648--2681. (Ah, so that's how the "insert citation" butto …

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 …