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Abstract
We prove the Shepp–Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof refines an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin’s original conjecture, to consider Rényi and Tsallis entropies.
Original language  English 

Pages (fromto)  36383649 
Number of pages  12 
Journal  Bernoulli 
Volume  23 
Issue number  4B 
Early online date  23 May 2017 
DOIs  
Publication status  Published  Nov 2017 
Keywords
 Bernoulli sums
 concavity
 entropy
 transportation of measure
 Poisson binomial distribution
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Profiles

Professor Oliver T Johnson
 School of Mathematics  Professor of Information Theory
 Statistical Science
 Probability, Analysis and Dynamics
 Probability
Person: Academic , Academic , Member