2012/07/26 by Varun Jog, Jog, Varun, Venkat Anantharam +1 · 1 citation
Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Group Theory (math.GR) #Information Theory (cs.IT) #Limits and Structures in Graph Theory #Probability (math.PR) #Wireless Communication Security Techniques
paper · pdf · doi:10.48550/arxiv.1207.6355
openalex publication_date 2012/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Shannon's Entropy Power Inequality can be viewed as characterizing the\nminimum differential entropy achievable by the sum of two independent random\nvariables with fixed differential entropies. The entropy power inequality has\nplayed a key role in resolving a number of problems in information theory. It\nis therefore interesting to examine the existence of a similar inequality for\ndiscrete random variables. In this paper we obtain an entropy power inequality\nfor random variables taking values in an abelian group of order 2n, i.e. for\nsuch a group G we explicitly characterize the function fG(x,y) giving the\nminimum entropy of the sum of two independent G-valued random variables with\nrespective entropies x and y. Random variables achieving the extremum in this\ninequality are thus the analogs of Gaussians in this case, and these are also\ndetermined. It turns out that fG(x,y) is convex in x for fixed y and, by\nsymmetry, convex in y for fixed x. This is a generalization to abelian groups\nof order 2n of the result known as Mrs. Gerber's Lemma.\n