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 $\mathbf{A}$ 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 $\mathbb{E}[\bar{X}(m)]$ and the variance $\mathrm{Var}[\bar{X}(m)]$ of $\bar{X}(m)$. Using the expression for $\mathrm{Var}[\bar{X}(m)]$, we establish some equalities and inequalities involving $\mathrm{Var}[\bar{X}(m)]$, the Frobenius norm, the largest eigenvalue, the largest singular value and the coherence of a matrix $\mathbf{A}$. 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 $\mathbf{A}$ 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.

arXiv comment12 pages, no figures; literature is extended and minor corrections in its citations have been made

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• Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
• Reconstruction of Sparse and Nonsparse Signals from a Reduced Set of Samples
• On some discrete random variables arising from recent study on statistical analysis of compressive sensing
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Cited by in corpus (4)

• On some discrete random variables arising from recent study on statistical analysis of compressive sensing
• Several generalizations and variations of Chu-Vandermonde identity
• A note on some sub-Gaussian random variables
• On some random variables involving Bernoulli random variable