A characterization for mixtures of semi-Markov processes.
type: Item
instance of: scholarly article
title: A characterization for mixtures of semi-Markov processes.
zbMATH Open document ID: 1033.60090
DOI: 10.1016/S0167-7152(02)00324-3
author: Lucia Ladelli
author: Ilenia Epifani
author: S. Fortini
published in: Statistics \& Probability Letters
publication date: 2003-05-07T00:00:00Z
review text: A stepped right-continuous random process with a countable space of states \(I\) is considerd. It can be represented by the random sequence \((\sigma_j, \xi_j)_1^\infty\), where \(\sigma_j\) is the \(j\)th jump time, and \(\xi_j\in I\) is the value of the process at time \(\sigma_j\). Let \(\nu_{im}\) be the \(m\)th hitting time of the state \(i\in I\), \(T_{im}= \sigma_{\nu_{im}+1}- \sigma_{\nu_{im}}\), and \(S_{im}= \xi_{\nu_{im}+1}\). The random matrix \((T,S) \equiv (T_{im}, S_{im})_{k\times n}\) \((i\in K\subset I\), \(k\equiv| K|\), \(1\leq m\leq n)\) is called row-wise partially exchangeable if its distribution is invariant with respect to the group of all permutations of the numbers \((1,\dots, n)\) of its columns. The random process is proved to have for any \(K\) and \(n\) a row-wise partially exchangeable matrix \((T,S)\) iff it is representable as a mixture of stepped semi-Markov processes on the space \(I\). The theorem is strengthened for the mixture of stepped Markov processes.
Mathematics Subject Classification ID: 60K15
Mathematics Subject Classification ID: 60G09
zbMATH DE Number: 1907021
zbMATH Keywords: holding times
zbMATH Keywords: Markov chains
zbMATH Keywords: Markov exchangeability
zbMATH Keywords: exchangeability
zbMATH Keywords: semi-Markov processes
zbMATH Keywords: semi-Markov representation
zbMATH Keywords: successor state
reviewed by: B. P. Harlamov
MaRDI profile type: Publication
cites work: De Finetti's theorem for Markov chains
cites work: Q3827377
cites work: Q5328484
cites work: On mixtures of distributions of Markov chains.
cites work: Invariants Under Mixing which Generalize de Finetti's Theorem
cites work: Invariants Under Mixing Which Generalize de Finetti's Theorem: Continuous Time Parameter
cites work: Q5614313
cites work: Q4255523
cites work: Characterizations and embedding properties in exchangeability
cites work: Markov Chains
cites work: Markov Renewal Processes: Definitions and Preliminary Properties
cites work: Characterizing Markov exchangeable sequences
cites work: Urn models for Markov exchangeability
cites work: A finite form of de Finetti's theorem for stationary Markov exchangeability
full work available at URL: s0167-7152(02)00324-3
OpenAlex ID: W2055857421
description: scientific article; zbMATH DE number 1907021