On some discrete random variables arising from recent study on statistical analysis of compressive sensing

Abstract

The recent paper [27] provides a statistical analysis for efficient detection of signal components when missing data samples are present. Here we focus our attention to some complex-valued discrete random variables $X_l(m,N)$ ($0\le l\le N-1$, $1\le M\le N$), which are closely related to the random variables investigated by LJ. Stankovi\'c, S. Stankovi\'c and M. Amin in \cite{ssa}. In particular, by using a combinatorial approach, we prove that for $l\not=0$ the expected value of $X_l(m,N)$ is equal to zero, and we deduce the expression for the variance of the random variables $X_l(m,N)$. The same results are also deduced for the real part $U_l(m,N)$ and the imaginary part $V_l(m,N)$ of $X_l(m,N)$, as well as the facts that the $k$th moments of $U_l(m,N)$ and $V_l(m,N)$ are equal to zero for every positive integer $k$ which is not divisible by $N/\gcd(N,l)$. Moreover, some additional assertions and examples concerning the random variables $X_l(m,N)$, $U_l(m,N)$ and $V_l(m,N)$ are also presented.