  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.
