Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

Abstract

We study the upper-lower class behavior of weighted sums ∑ k=1 n a k X k , where X k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.