A generalization of some random variables involving in certain compressive sensing problems

Abstract

In this paper we give a generalization of the discrete complex-valued random variable defined and investigated in \cite{ssa} and \cite{m8}. We prove the statements concerning the expressions for the excepted value and the variance of this random variable. In partucular, such a random variable here is defined for each of $m$ rows of any $m\times N$ complex or real matrix ${\rm{\bf A}}$ with $1\le m\le N$. We consider the arithmetic mean $\bar{X}(m)$ of these $m$ random variables and we deduce the expressions for the expected value $\Bbb E[\bar{X}(m)]$ and the variance ${\rm{ Var}}[\bar{X}(m)]$ of $\bar{X}(m)$. Using the expression for ${\rm{ Var}}[\bar{X}(m)]$, we establish some equalities and inequalities involving ${\rm{Var}}[\bar{X}(m)]$, the Frobenius norm, the largest eigenvalue, the largest singular value and the coherence of a matrix ${\rm{\bf A}}$. It is showed that some of these estimates are closely related to the Welch bound of the coherence of a $m\times N$ complex or real matrix ${\rm{\bf A}}$ with $1\le m\le N$. Taking into account that the value of coherence of the measurement matrix in the theory of compressive sensing has a significant role, we believe that our results should be useful for some topics of this theory.