A strong limit theorem under no assumption of independence, stationarity or various dependences
type: Item
instance of: scholarly article
title: A strong limit theorem under no assumption of independence, stationarity or various dependences
zbMATH Open document ID: 0810.60031
DOI: 10.1016/0167-7152(94)90224-0
published in: Statistics \& Probability Letters
publication date: 1994-11-06T00:00:00Z
review text: The following theorem is proved. Let \((q_ n, n \geq 0)\) be a sequence of positive integers, \(I_ n = \{0,1, \dots, q_ n\}\), \((X_ n, n \geq 0)\) be a sequence of random variables, where \(X_ n\) takes on values in \(I_ n\), and \[ P(X_ 0 = x_ 0, \dots, X_ n = x_ n) = p(x_ 0, \dots, x_ n) > 0 \quad \text{for all} \quad x_ i \in I_ i,\;0 \leq i \leq n. \] If \(\sum^ \infty_{n=1} q^ 2_ n/n^ 2 < \infty\), then \[ \lim_ n {1 \over n} \sum^ n_{i=1} \biggl( X_ i - E \bigl( X_ i \mid X_ 0, \dots, X_{i-1} \bigr) \biggr) = 0 \quad \text{a.s.}. \]
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 681890
zbMATH Keywords: strong limit theorem
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: conditional expectation
author: Wen Liu
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)90224-0
OpenAlex ID: W2059948758
cites work: Relative entropy densities and a class of limit theorems of the sequence of m-valued random variables
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description: scientific article; zbMATH DE number 681890