Abstract. Let $\{X_n\} \subset L_p({\bf P})$, $1< p\le2$, $q=p/(p-1)$, be a sequence of martingale differences. We prove that the Komlós--Révész type weighted averages ${\sum_{k=1}^n (X_k/\|X_k\|_p^q)\over\sum _{k=1}^n (1/\|X_k\|_p^q)}$ converge a.s. and in the $L_p$-norm, and the limit is $0$ if and only if $\sum_{n=1}^\infty(1/\|X_n\|_p^q)=\infty $. We show also that convergence need not hold when we deal with a centered uncorrelated sequence (whether the series $\sum_{n=1}^\infty(1/\|X_n\|_2^2)$ converges or not). Furthermore, for $1< p< 2$ all the results of Komlós--Révész are extended to symmetric independent $p$-stable random variables.

AMS Subject Classification (1991): 60F15, 60F25; 60G42, 60G52, 62F12

Keyword(s): independent random variables, martingale differences, p, -stable random variables, weighted averages, a.s. convergence, norm convergence, consistent estimation of a common mean

Received November 8, 2007, and in revised form March 6, 2008. (Registered under 6053/2009.)