Electronic Journal of Probability

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

Kouhei Uchiyama

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Abstract

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

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

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

Permanent link to this document
http://projecteuclid.org/euclid.ejp/1464818477

Digital Object Identifier
doi:10.1214/EJP.v12-396

Mathematical Reviews number (MathSciNet)
MR2299915

Zentralblatt MATH identifier
1134.60055

Subjects
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

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

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

Citation

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.


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