A Proof of a Conjecture of Bobkov and Houdré

Abstract

S. G. Bobkov and C. Houdré recently posed the following question on the Internet (Problem posed in Stochastic Analysis Digest no. 15 (9/15/1995)): Let $X,Y$ be symmetric i.i.d. random variables such that $$P(|X+Y|/2 \geq t) \leq P(|X| \geq t),$$ for each $t>0$. Does it follow that $X$ has finite second moment (which then easily implies that $X$ is Gaussian)? In this note we give an affirmative answer to this problem and present a proof. Using a dierent method K. Oleszkiewicz has found another proof of this conjecture, as well as further related results.