Simple Extension of Halász's Result

Abstract

Inspired by the idea of Bernoulli decomposition, we give a simple proof for a generalization of Halász anti--concentration result about random sum of vectores in ${\mathbb{R}}^d$. From our results, we can give one upper bound for the probability of a random circulant matrix ${\mathcal{C}}_n$ with independent positive (or negative) random variables is singular, in fact, we prove $\mathbb{P}({\mathcal{C}}_n\mathrm{is}\mathrm{singular})\le C_{1,F}n(\varphi (n){)}^{-3}$, where the constant $C_{1,F}>0$ depends on the distribution of the entries and $\varphi (\cdot )$ is the Euler totient function. Also, if ${\mathcal{SC}}_n$ is a random symmetric circulant matrix with independent positive (or negative) integer random variable entries, we show $\mathbb{P}({\mathcal{SC}}_n\mathrm{is}\mathrm{singular})\le C_{2,F}n(\varphi (n){)}^{-3/2}$, where the constant $C_{2,F}>0$ depends on the distribution of the entries. It is possible to assume that the entries of a random circulant matrix are not identically distributed.

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References in corpus (2)

• On the g-Circulant Matrix involving the Generalized k-Horadam Numbers
• Salem-Zygmund Inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices