On the Markov renewal theorem

Gerold Alsmeyer

Stochastic Processes and their Applications, 1994, vol. 50, issue 1, 37-56

Abstract: Let (S, £) be a measurable space with countably generated [sigma]-field £ and (Mn, Xn)n[greater-or-equal, slanted]0 a Markov chain with state space S x and transition kernel :S x ( [circle times operator] )-->[0, 1]. Then (Mn,Sn)n[greater-or-equal, slanted]0, where Sn = X0+...+Xn for n[greater-or-equal, slanted]0, is called the associated Markov random walk. Markov renewal theory deals with the asymptotic behavior of suitable functionals of (Mn,Sn)n[greater-or-equal, slanted]0 like the Markov renewal measure [Sigma]n[greater-or-equal, slanted]0P((Mn,Sn)[epsilon]Ax (t+B)) as t-->[infinity] where A[epsilon] and B denotes a Borel subset of . It is shown that the Markov renewal theorem as well as a related ergodic theorem for semi-Markov processes hold true if only Harris recurrence of (Mn)n[greater-or-equal, slanted]0 is assumed. This was proved by purely analytical methods by Shurenkov [15] in the one-sided case where (x,Sx[0,[infinity])) = 1 for all x[epsilon]S. Our proof uses probabilistic arguments, notably the construction of regeneration epochs for (Mn)n[greater-or-equal, slanted]0 such that (Mn,Xn)n[greater-or-equal, slanted]0 is at least nearly regenerative and an extension of Blackwell's renewal theorem to certain random walks with stationary, 1-dependent increments.

Keywords: Markov; renewal; theory; Markov; random; walk; semi-Markov; process; Harris; recurrence>; (null); regeneration; epochs; Blackwell's; renewal; theorem; random; walks; with; stationary; 1-dependent; increments; coupling (search for similar items in EconPapers)
Date: 1994
References: Add references at CitEc
Citations: View citations in EconPapers (11)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/0304-4149(94)90146-5
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:50:y:1994:i:1:p:37-56

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().