On the Maximum Entropy of a Sum of Independent Discrete Random Variables

08/03/2020
by   Mladen Kovačević, et al.
0

Let X_1, …, X_n be independent random variables taking values in the alphabet {0, 1, …, r}, and S_n = ∑_i = 1^n X_i. The Shepp–Olkin theorem states that, in the binary case (r = 1), the Shannon entropy of S_n is maximized when all the X_i's are uniformly distributed, i.e., Bernoulli(1/2). In an attempt to generalize this theorem to arbitrary finite alphabets, we obtain a lower bound on the maximum entropy of S_n and prove that it is tight in several special cases. In addition to these special cases, an argument is presented supporting the conjecture that the bound represents the optimal value for all n, r, i.e., that H(S_n) is maximized when X_1, …, X_n-1 are uniformly distributed over {0, r}, while the probability mass function of X_n is a mixture (with explicitly defined non-zero weights) of the uniform distributions over {0, r} and {1, …, r-1}.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
08/13/2018

On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs

Suppose that you have n colours and m mutually independent dice, each of...
research
06/01/2021

Entropy and the Discrete Central Limit Theorem

A strengthened version of the central limit theorem for discrete random ...
research
09/02/2019

A Tight Uniform Continuity Bound for Equivocation

We prove a tight uniform continuity bound for the conditional Shannon en...
research
07/19/2018

On Chebotarëv's nonvanishing minors theorem and the Biró-Meshulam-Tao discrete uncertainty principle

Chebotarëv's theorem says that every minor of a discrete Fourier matrix ...
research
04/24/2018

Rate-Distortion Theory for General Sets and Measures

This paper is concerned with a rate-distortion theory for sequences of i...
research
08/13/2020

Infinite Divisibility of Information

We study an information analogue of infinitely divisible probability dis...
research
05/04/2023

Chain Rules for Renyi Information Combining

Bounds on information combining are a fundamental tool in coding theory,...

Please sign up or login with your details

Forgot password? Click here to reset