Strong law of large numbers for sums of products

Abstract

Let $X, X_n, n \ge 1$, be a sequence of independent identically distributed random variables. We give necessary and sufficient conditions for the strong law of large numbers

n^{-k/p} \sum_{1\lei_1 \le i_2 <\dots < i_k \le n} X_{i_1}X_{i_2}\dots X_{i_k} \to 0\quad\text{a.s.}

for $k =2$ without regularity conditions on $X$, for $k \geq 3$ in three cases: (i) symmetric X, (ii) $P \{X \leq 0\} =1 and (iii) regularly varying $P\{|X|}> x\}$ as $x \to \infty$, without further conditions, and for general X and k under a condition on the growth of the truncated mean of X. Randomized, centered, squared and decoupled strong laws and general normalizing sequences are also considered.