Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T13:17:47.549Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasistochastic matrices and Markov renewal theory

Part of: Markov processes Special processes

Published online by Cambridge University Press:  30 March 2016

Gerold Alsmeyer
Gerold Alsmeyer*
Affiliation:
Institute of Mathematical Statistics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, D-48149 Münster, Germany. Email address: gerolda@math.uni-muenster.de.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)i,j∈𝓈, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑n≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Keywords

Quasistochastic matrixMarkov random walkMarkov renewal equationMarkov renewal theoremspread outStone-type decompositionage-dependent multitype branching processrandom difference equationperpetuity

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J45: Probabilistic potential theory 60K15: Markov renewal processes, semi-Markov processes
Type
Part 8. Markov processes and renewal theory
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 359 - 376
Copyright
Copyright © Applied Probability Trust 2014 

References

Alsmeyer, G., and Mentemeier, S. (2012). Tail behaviour of stationary solutions of random difference equations: the case of regular matrices. J. Difference Equat. Appl./ 18, 1305–1332.Google Scholar
Alsmeyer, G., Iksanov, A. and Rösler, U. (2009). On distributional properties of perpetuities. J. Theoret. Prob./ 22, 666–682.Google Scholar
Asmussen, S. (1989). Aspects of matrix Wiener-Hopf factorisation in applied probability. Math. Sci./ 14, 101–116.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Athreya, K. B., and Rama Murthy, K. (1976). Feller's renewal theorem for systems of renewal equations. J. Indian Inst. Sci./ 58, 437–459.Google Scholar
Athreya, K. B., McDonald, D., and Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Prob./ 6, 788–797.Google Scholar
Brandt, A. (1986). The stochastic equation {Yn+1=AnYn+Bn# with stationary coefficients. Adv. Appl. Prob./ 18, 211–220.Google Scholar
Buraczewski, D. et al.(2009). Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Prob. Theory Relat. Fields 145, 385–420.Google Scholar
Cinlar, E. (1969). Markov renewal theory. Adv. Appl. Prob./ 1, 123–187.Google Scholar
Cinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Crump, K. S. (1970). On systems of renewal equations. J. Math. Anal. Appl./ 30, 425–434.Google Scholar
De Saporta, B. (2003). Renewal theorem for a system of renewal equations. Ann. Inst. H. Poincaré Prob. Statist./ 39, 823–838.Google Scholar
De Saporta, B. (2005). Tail of the stationary solution of the stochastic equation Yn+1=anYn+bn# with Markovian coefficients. Stoch. Process. Appl./ 115, 1954–1978.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob./ 1, 126–166.Google Scholar
Goldie, C. M., and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob./ 28, 1195–1218.Google Scholar
Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math./ 131, 207–248.Google Scholar
Le Page, É. (1983). Théorèmes de renouvellement pour les produits de matrices aléatoires. Équations aux différences aléatoires. In Séminaires de Probabilités Rennes 1983, 116pp.Google Scholar
Mode, C. J. (1971). Multitype age-dependent branching processes and cell cycle analysis. Math. Biosci./ 10, 177–190.Google Scholar
Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications (Modern Anal. Comput. Meth. Sci. Math. 34). American Elsevier Publishing, New York.Google Scholar
Niemi, S., and Nummelin, E. (1986). On nonsingular renewal kernels with an application to a semigroup of transition kernels. Stoch. Process. Appl./ 22, 177–202.Google Scholar
Roitershtein, A. (2007). One-dimensional linear recursions with Markov-dependent coefficients. Ann. Appl. Prob./ 17, 572–608.Google Scholar
Roitershtein, A., and Zhong, Z. (2013). On random coefficient {INAR}(1) processes. Sci. China Math./ 56, 177–200.Google Scholar
Seneta, E. (2006). Non-Negative Matrices and Markov Chains. Springer, New York.Google Scholar
Sgibnev, M. S. (2001). Stone's decomposition for the matrix renewal measure on a half-axis. Mat. Sb./ 192, 97–106.Google Scholar
Sgibnev, M. S. (2006). A matrix analogue of the Blackwell renewal theorem on the line. Mat. Sb./ 197, 69–86.Google Scholar
Sgibnev, M. S. (2010). On the uniqueness of the solution of a system of renewal-type integral equations on the line. Izv. Ross. Akad. Nauk Ser. Mat. 74, 157–168 (in Russian). English translation: Izv. Math. 74, 595–606.Google Scholar
Shurenkov, V. M. (1984). On Markov renewal theory. Theory Prob. Appl./ 29, 247–265.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob./ 11, 750–783.CrossRefGoogle Scholar