Sur la convergence presque sure, au sens de Cesaro d'ordre α, 0<α<1, de variables aléatoires indepéndantes et identiquement distribuées

Abstract

We prove the following theorem: «Given 0<α≦1, the (C, α)-means of a sequence of i.i.d. random variables X n converge a.s. iff E|X n|1/α<∞.» For 1/2<α≦1 and 0<α<1/2 this result is essentially known. We give here a proof of the case α=1/2; an important tool is a theorem of Hsu and Robbins [5].