Laws of large numbers for quadratic forms, maxima of products and truncated sums of i.i.d. random variables
type: Item
instance of: scholarly article
title: Laws of large numbers for quadratic forms, maxima of products and truncated sums of i.i.d. random variables
zbMATH Open document ID: 0833.60030
author: Jack Cuzick
author: Joel Zinn
author: Evarist Giné
published in: The Annals of Probability
publication date: 1996-03-11T00:00:00Z
review text: Let \(X_i\) be square-integrable i.i.d. random variables. Necessary and sufficient conditions are given for \({1\over \gamma_n} \max_{1 \leq i < j \leq n} |X_i X_j|\to 0\) a.s. in general and for \({1\over \gamma_n} \sum_{1 \leq i \neq j \leq n} X_i X_j \to 0\) a.s. under some restrictions. Strong laws are obtained for \(X_{1:n} X_{k:n}\), where \(X_{j:n}\) is the \(j\)th largest absolute value among \(X_1, \dots, X_n\). Also, sharp asymptotic bounds are given for \({1\over b_n} \sum^n_{i = 1} X_i \mathbf{ {1}}_{|X_i|< b_n}\) for relatively small \(b_n\).
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 796544
zbMATH Keywords: quadratic forms of random variables
zbMATH Keywords: maxima of products
zbMATH Keywords: truncated sums
zbMATH Keywords: strong laws
zbMATH Keywords: asymptotic bounds
reviewed by: Heinrich Hering
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988388
OpenAlex ID: W1997849008
DOI: 10.1214/AOP/1176988388
description: scientific article; zbMATH DE number 796544