Approximate Discrete Entropy Monotonicity for Log-Concave Sums
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It is proved that for any n ≥ 1, if X_1,…,X_n are i.i.d. integer-valued, log-concave random variables then h(X_1+…+X_n + U_1+…+U_n) = H(X_1+…+X_n) + o(1), as H(X_1) →∞, where h stands for the differential entropy, H dentoes the (discrete) Shannon entropy and U_1,…,U_n are independent continuous uniforms on (0,1). As a corollary, it is shown that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: H(X_1+…+X_n+1) ≥ H(X_1+⋯+X_n) + 1/2log(n+1/n) - o(1) as H(X_1) →∞. Explicit bounds for the o(1)-terms are provided.
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