Strong law of large numbers for multilinear forms. (English) Zbl 0937.60016

Gadidov, Anda

Ann. Probab. 26, No. 2, 902-923 (1998).

Summary: Let \(m\geq 2\) be a nonnegative integer and let \(\{X^{(l)},X^{(l)}_i\}_{i\in\mathbb{N}}\), \(l=1,\dots,m\), be \(m\) independent sequences of independent and identically distributed symmetric random variables. Define \(S_n=\sum_{1\leq i_1,\dots,i_m\leq n}X_{i_1}^{(1)}\cdots X_{i_m}^{(m)}\), and let \(\{\gamma_n\}_{n\in\mathbb{N}}\) be a nondecreasing sequence of positive numbers, tending to infinity and satisfying some regularity conditions. For \(m=2\) necessary and sufficient conditions are obtained for the strong law of large numbers \(\gamma_n^{-1}S_n\to 0\) a.s. to hold, and for \(m>2\) the strong law of large numbers is obtained under a condition on the growth of the truncated variance of the \(X^{(l)}\).

MSC:

60F15

Strong limit theorems

Keywords:

strong laws; multilinear forms; \(U\)-statistics; martingale; maximal inequality

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