On some random variables involving Bernoulli random variable

Abstract

Motivated by the recent investigations given in [25] and the fact that Bernoulli probability-type models were often used in the study on some problems in theory of compressive sensing, here we define and study the complex-valued discrete random variables $\widetilde{X}_l(m,N)$ ($0\le l\le N-1$, $1\le m\le N$). Each of these random variables is defined as a linear combination of $N$ independent identically distributed $0-1$ Bernoulli random variables. We prove that for $l\not=0$, $\widetilde{X}_l(m,N)$ is the zero-mean random variable, and we also determine the variance of $\widetilde{X}_l(m,N)$ and its real and imaginary parts. Notice that $\widetilde{X}_l(m,N)$ belongs to the class of sub-Gaussian random variables that are significant in some areas of theory of compressive sensing. In particular, we prove some probability estimates for the mentioned random variables. These estimates are used to establish the upper bounds of the sub-Gaussian norm of their real and imaginary parts. We believe that our results should be implemented in certain applications of sub-Gaussian random variables for solving some problems in compressive sensing of sparse signals.