Solution of Shannon’s problem on the monotonicity of entropy

Abstract

It is shown that if X 1 , X 2 , … X_1,X_2,\ldots are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum \[ Ent ⁡ ( X 1 + ⋯ + X n n ) \operatorname {Ent} \left (\frac {X_{1}+\cdots + X_{n}}{\sqrt {n}} \right ) \] is an increasing function of n n . The result also has a version for non-identically distributed random variables or random vectors.