Electronic Journal of Probability

Asymptotic Estimates Of The Green Functions And Transition Probabilities For Markov Additive Processes

Kouhei Uchiyama

Full-text: Open access


In this paper we shall derive asymptotic expansions of the Green function and the transition probabilities of Markov additive (MA) processes $(\xi_n, S_n)$ whose first component satisfies Doeblin's condition and the second one takes valued in $Z^d$. The derivation is based on a certain perturbation argument that has been used in previous works in the same context. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important for instance in computation of the harmonic measures of a half space for certain models. We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be clarified and it will be shown that in a simple manner the process, if not degenerate, are transformed to another one that satisfies Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if the MA processes is irreducible as a Markov process, then the Green function is expanded quite similarly to that of a classical random walk on $Z^d$.

Article information

Electron. J. Probab. Volume 12 (2007), paper no. 6, 138-180.

Accepted: 16 February 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J05: Discrete-time Markov processes on general state spaces

asymptotic expansion harmonic analysis semi-Markov process random walk with internal states perturbation aperiodicity ergodic Doeblin's condition

This work is licensed under a Creative Commons Attribution 3.0 License.


Uchiyama, Kouhei. Asymptotic Estimates Of The Green Functions And Transition Probabilities For Markov Additive Processes. Electron. J. Probab. 12 (2007), paper no. 6, 138--180. doi:10.1214/EJP.v12-396. http://projecteuclid.org/euclid.ejp/1464818477.

Export citation


  • Babillot, M. Théorie du renouvellement pour des chaînes semi-markoviennes transientes. (French) [Renewal theory for transient semi-Markov chains] Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), no. 4, 507–569.
  • Çinlar, Erhan. Markov additive processes. I, II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), 85–93; ibid. 24 (1972), 95–121. (48 #7389)
  • Doob, J. L. Stochastic processes. John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp. (15,445b)
  • Fukai, Yasunari. Hitting distribution to a quadrant of two-dimensional random walk. Kodai Math. J. 23 (2000), no. 1, 35–80.
  • Fukai, Yasunari; Uchiyama, Kôhei. Potential kernel for two-dimensional random walk. Ann. Probab. 24 (1996), no. 4, 1979–1992. Y. Guivarc'h, Application d'un theoreme limite local a la transience et a la recurrence des marches de Markov, Lecture Notes, 1096, 301-332, Springer-Verlag, 1984.
  • Itô, Kiyosi; McKean, H. P., Jr. Potentials and the random walk. Illinois J. Math. 4 1960 119–132. (22 #12317)
  • Gnedenko, B. V.; Kolmogorov, A. N. Limit distributions for sums of independent random variables. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. ix+264 pp. (16,52d)
  • Höglund, Thomas. A multi-dimensional renewal theorem for finite Markov chains. Ark. Mat. 28 (1990), no. 2, 273–287.
  • Hara, Takashi; van der Hofstad, Remco; Slade, Gordon. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 (2003), no. 1, 349–408.
  • Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman. Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp. (48 #1287) T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Heidelberg-New York, 1980.
  • Kesten, Harry. Renewal theory for functionals of a Markov chain with general state space. Ann. Probability 2 (1974), 355–386. (51 #1992)
  • Kesten, Harry. Hitting probabilities of random walks on $Zsp d$. Stochastic Process. Appl. 25 (1987), no. 2, 165–184.
  • Kesten, Harry. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), 207–248. (55 #13595) T. Kazami and K. Uchiyama, Random walks on periodic graphs. (to appear in TAMS)
  • Keilson, J.; Wishart, D. M. G. A central limit theorem for processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 60 1964 547–567. (29 #6523)
  • Krámli, András; Szász, Domokos. Random walks with internal degrees of freedom. I. Local limit theorems. Z. Wahrsch. Verw. Gebiete 63 (1983), no. 1, 85–95.
  • Lawler, Gregory F.; Limic, Vlada. The Beurling estimate for a class of random walks. Electron. J. Probab. 9 (2004), no. 27, 846–861 (electronic).
  • Nagaev, S. V. Some limit theorems for stationary Markov chains. (Russian) Teor. Veroyatnost. i Primenen. 2 1957 389–416. (20 #1355)
  • Ney, P.; Nummelin, E. Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 (1987), no. 2, 561–592.
  • Spitzer, Frank. Principles of random walk. The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1964 xi+406 pp. (30 #1521)
  • Stein, Elias M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp. (44 #7280)
  • Takenami, Toshiyuki. Local limit theorem for random walk in periodic environment. Osaka J. Math. 39 (2002), no. 4, 867–895.
  • Uchiyama, Kôhei. Green's functions for random walks on $Zsp N$. Proc. London Math. Soc. (3) 77 (1998), no. 1, 215–240.