Simple Extension of Hal\'asz's Result

Abstract

Inspired by the idea of Bernoulli decomposition, we give a simple proof for a generalization of Hal\'asz 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 $\mathbb{P}\ (\mathcal{C}_n \mathrm{\ is\ singular}) \leq C_{1,F} n(\phi(n))^{-3}$, where the constant $C_{1,F}>0$ depends on the distribution of the entries and $\phi(\cdot)$ 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 $\mathbb{P}\ (\mathcal{SC}_n \mathrm{\ is\ singular})\leq C_{2,F} n(\phi(n))^{-3/2}$, 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.