Strong laws for weighted sums of independent identically distributed random variables
type: Item
instance of: scholarly article
title: Strong laws for weighted sums of independent identically distributed random variables
zbMATH Open document ID: 0883.60023
DOI: 10.1215/S0012-7094-97-08808-6
published in: Duke Mathematical Journal
publication date: 1998-02-23T00:00:00Z
review text: The following multidimensional form of Kolmogorov's strong law of large numbers is established. Theorem 1. Let \(H\) be a positive integer. Given an i.i.d. sequence \((X_{1,n})_n\) on the probability measure space \((\Omega_1, F_1, \mu_1)\) satisfying \(E(|X_{1,1}|)<\infty\), it is possible to find a set of full measure \(\widetilde\Omega_1\) such that if \(x_1\in \widetilde\Omega_1\), the following holds: (1) For all probability measure spaces \((\Omega_2, F_2, \mu_2)\) and all i.i.d. sequences \((X_{2,n})_n\) such that \(E(|X_{2,1}|)<\infty\), it is possible to find a set of full measure \(\widetilde\Omega_2\) such that if \(x_2\in\widetilde\Omega_2\) the following holds: (2) \dots \dots (\(H-1\)) For all probability measure spaces \((\Omega_H, F_H, \mu_H)\) and all i.i.d. sequences \((X_{H,n})_n\) such that \(E(|X_{H,1}|)<\infty\), it is possible to find a set of full measure \(\widetilde\Omega_H\) for which, if \(x_H\in\widetilde\Omega_H\) we have \[ \frac{1}{N}\sum_{n=1}^{N} X_{1,n}(x_1)X_{2,n}(x_2)\cdots X_{H,n}(x_H)\to \prod_{i=1}^H E(X_{i,1}) \qquad\text{as}\quad N\to\infty. \] The main difficulty in the proof of this theorem is that at each stage the null set does not depend on the incoming i.i.d. sequences and associated probability spaces. Further ergodic dynamical systems are considered.
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 1042481
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: row-wise independent
zbMATH Keywords: ergodic dynamical system
zbMATH Keywords: maximal inequality
author: Idris Assani
reviewed by: Andrei Volodin
MaRDI profile type: Publication
cites work: Pointwise ergodic theorems for arithmetic sets. With an appendix on return-time sequences, jointly with Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein
cites work: Complete Convergence and the Law of Large Numbers
cites work: Convergence of weighted averages of independent random variables
cites work: Inequalities for the ergodic maximal function
cites work: Maximal inequalities of weak type
cites work: Q4084474
full work available at URL: s0012-7094-97-08808-6
OpenAlex ID: W60076420
description: scientific article; zbMATH DE number 1042481