The Annals of Probability

Strong law of large numbers for sums of products

Cun-Hui Zhang

Full-text: Open access

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

Keywords
Strong law of large numbers Marcinkiewicz–Zygmund law U-statistics quadratic forms decoupling maximum of products

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


Export citation