A limit property of random conditional probabilities
type: Item
instance of: scholarly article
title: A limit property of random conditional probabilities
zbMATH Open document ID: 0964.60032
DOI: 10.1016/S0167-7152(00)00061-4
author: Wen Liu
published in: Statistics \& Probability Letters
publication date: 2001-07-08T00:00:00Z
review text: This paper provides a strong law of large numbers for the harmonic mean of random conditional probabilities \(\{p_k(X_k\mid X_0,\dots, X_{k-1}): k= 1,\dots, n\}\), where \(p_k(x_k\mid x_0,\dots, x_{k-1})= P(X_k= x_k\mid X_0= x_0,\dots, X_{k-1}= x_{k-1})\), \(k\geq 1\), and \(\{X_n:n\geq 0\}\) is a sequence of random variables taking values in \(S= \{1,\dots, N\}\). The proof makes use of conditional moment generating functions and differentiation on a net. As an application, almost sure limiting properties of the arithmetic mean of the (random) transition probabilities in an \(m\)th order nonhomogeneous Markov chain are discussed.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 1517226
zbMATH Keywords: random conditional probability
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: harmonic mean
zbMATH Keywords: conditional moment generating function
zbMATH Keywords: Markov chain
zbMATH Keywords: transition probabilities
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
cites work: Some strong limit theorems relative to the geometric average of random transition probabilities of arbitrary finite nonhomogeneous Markov chains
cites work: An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains
cites work: Analytic Inequalities
full work available at URL: s0167-7152(00)00061-4
OpenAlex ID: W2080568826
description: scientific article; zbMATH DE number 1517226