## The Annals of Probability

### Strong law of large numbers for sums of products

Cun-Hui Zhang

#### 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.

#### Article information

Source
Ann. Probab., Volume 24, Number 3 (1996), 1589-1615.

Dates
First available in Project Euclid: 9 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1065725194

Digital Object Identifier
doi:10.1214/aop/1065725194

Mathematical Reviews number (MathSciNet)
MR1411507

Zentralblatt MATH identifier
0868.60024

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G50: Sums of independent random variables; random walks

#### Citation

Zhang, Cun-Hui. Strong law of large numbers for sums of products. Ann. Probab. 24 (1996), no. 3, 1589--1615. doi:10.1214/aop/1065725194. https://projecteuclid.org/euclid.aop/1065725194