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.
Citation
Cun-Hui Zhang. "Strong law of large numbers for sums of products." Ann. Probab. 24 (3) 1589 - 1615, July 1996. https://doi.org/10.1214/aop/1065725194
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