Markov additive processes. I: Eigenvalue properties and limit theorems
type: Item
instance of: scholarly article
title: Markov additive processes. I: Eigenvalue properties and limit theorems
zbMATH Open document ID: 0625.60027
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: Markov additive processes are of the form \(\{(X_ n,S_ n):n=0,1,2,...\}\) where \(X_ n\) is a Markov chain on \({\mathbb{E}}\) (with \(\sigma\)-field \({\mathcal E})\) and \(S_ n=\sum^{n}_{i=1}\xi_ i\) \((\in {\mathbb{R}}^ d)\), and the distribution of \((X_{n+1},\xi_{n+1})\) depends on the past only through \(X_ n\). Let \(I(\cdot)\) be the indicator function and \(E_ x\) denote the expectation conditional on \(X_ 0=x.\) In this paper a careful study is made of the eigenvalues and eigenvectors of the kernel defined for \((x,A)\in {\mathbb{E}}\times {\mathcal E}\) by \(\hat P(\alpha) = E_ x(e^{\alpha \xi}I(X_ 1\in A))\) for \(\alpha \in {\mathbb{R}}^ d\). This leads rather naturally to large deviation results for the process. These give information on the large deviation probabilities \[ n^{-1} \log (E_ xI(X_ n\in A,\quad n^{-1} S_ n\in \Gamma)) \] for large n and certain \(\Gamma\). These results are obtained under the condition that a certain multidimensional Laplace transform has an open set as its domain of finiteness. The sequel, part II, see the following title, Zbl 0625.60028, extends these results using a truncation argument and hence establishes that Markov additive processes obey a large deviation principle. In particular, the lower bound on the large deviation probabilities is obtained under very weak hypotheses.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60K15
zbMATH DE Number: 4015807
zbMATH Keywords: Markov additive processes
zbMATH Keywords: large deviation
author: Peter Ney
author: Esa Nummelin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992159
OpenAlex ID: W2090937733
DOI: 10.1214/AOP/1176992159
Recommended article: Q3794980
Recommended article: Large deviations of uniformly recurrent Markov additive processes
Recommended article: Q5320839
Recommended article: Large deviations for additive functionals of Markov chains
Recommended article: Q3352226
Recommended article: Q3843101
Recommended article: Q3409967
Recommended article: On fluid model and averaged parameters model for Markov additive processes with finite environment's state space
Recommended article: Large deviation principle for additive functionals of semi-Markov processes
Recommended article: Q4322887
description: scientific article; zbMATH DE number 4015807