Limit theorems for Markov chains of finite rank
type: Item
instance of: scholarly article
title: Limit theorems for Markov chains of finite rank
zbMATH Open document ID: 0548.60068
DOI: 10.1016/0024-3795(84)90071-5
author: A. H. Hoekstra
author: Fred W. Steutel
published in: Linear Algebra and its Applications
publication date: 1984-01-01T00:00:00Z
full work available at URL: limit-theorems-for-markov-chains-of-finite-rank(306b87a6-40c2-4cf8-9616-981dd7a3c1a2).html
review text: The authors consider a Markov chain \(\{X_ n\}\) on the real line, with a transition distribution function of the form \(P(X_ n\leq y| X_{n- 1}=x)=\sum^{r}_{1}a_ j(x)B_ j(y)\) for real values \(a_ j(x)\) and measures \(B_ j(y)\). Limit theorems for \(X_ n\), for \(X_ 1+...+X_ n\) and for \(\max (X_ 1,...,X_ n)\) are derived. The simplicity of analysis of this model, using matrix theory and the spectral decomposition, is stressed. This relies on such results as for example, that \(P(X_ n\in E| X_ 0=x)=a^ T(x)C^{n-1}B(E)\) where \(a^ T=(a_ 1,...,a_ n)\), \(B^ T=(B_ 1,...,B_ n)\), and C is the matrix with elements \(C_{jk}=\int a_ k(x)B_ j(dx)\).
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3874357
zbMATH Keywords: matrix theory
zbMATH Keywords: spectral decomposition
MaRDI profile type: Publication
OpenAlex ID: W2106847690
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description: scientific article; zbMATH DE number 3874357