Abstract
The complete representation of the Martin compactification for reflected random walks on a half-space \({\mathbb{Z}^d\times\mathbb{N}}\) is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the “radial” compactification obtained by Ney and Spitzer for the homogeneous random walks in \({\mathbb{Z}^d}\): convergence of a sequence of points \({z_n\in\mathbb{Z}^{d-1}\times\mathbb{N}}\) to a point of on the Martin boundary does not imply convergence of the sequence z n /|z n | on the unit sphere S d. Our approach relies on the large deviation properties of the scaled processes and uses Pascal’s method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.
References
Billingsley P.: Convergence of Probability Measures, Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1968)
Blinovskiĭ, V.M., Dobrushin, R.L.: Process level large deviations for a class of piecewise homogeneous random walks. The Dynkin Festschrift Progr. Probab., vol. 34, pp. 1–59. Birkhäuser, Boston (1994)
Borovkov A.A.: On Cramer transform, large deviations in boundary-value problems and conditional invariance principle. Sib. Math. J. 3, 493–509 (1995)
Borovkov A.A., Mogulskii A.A.: Large deviations for Markov chains in the positive quadrant. Russ. Math. Surv. 56(5), 803–916 (2001)
Dembo A., Zeitouni O.: Large Deviations Techniques and Applications. Springer, New York (1998)
Foley RD., McDonald DR.: Bridges and networks: exact asymptotics. Ann. Appl. Probab. 15(1B), 542–586 (2005)
Doob, J.L., Snell, J.L., Williamson, R.E.: Application of boundary theory to sums of independent random variables. In: Contributions to Probability and Statistics, pp. 182–197. Stanford University Press, Stanford (1960)
Dupuis P., Ellis R., Weiss A.: Large deviations for Markov processes with discontinuous statistics I: General upper bounds. Ann. Prob. 19(3), 1280–1297 (1991)
Dupuis P., Ellis R.S.: Large deviations for Markov processes with discontinuous statistics. II. Random walks. Prob. Theory Relat. Fields 91(2), 153–194 (1992)
Dupuis P., Ellis R.S.: The large deviation principle for a general class of queueing systems. I. Trans. Am. Math. Soc. 347(8), 2689–2751 (1995)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York (1998) Translated from the 1979 Russian original by Joseph Szücs
Hennequin P.L.: Processus de Markoff en cascade. Ann. Inst. H. Poincaré 18(2), 109–196 (1963)
Ignatiouk-Robert I.: Sample path large deviations and convergence parameters. Ann. Appl. Prob. 11(4), 1292–1329 (2001)
Ignatiouk-Robert, I.: Large deviations for processes with discontinuous statistics, pp. 1479–1508 (2005)
Ignatiouk-Robert I.: Martin boundary of a killed random walk on a half-space. J. Theor. Probab. 21(1), 35–68 (2008)
Ignatyuk, I.A., Malyshev, V.A., Shcherbakov, V.V.: The influence of boundaries in problems on large deviations. Uspekhi Matematicheskikh Nauk 49, no. 2(296), 43–102 (1994)
Kurkova I.A., Malyshev V.A.: Martin boundary and elliptic curves. Markov Process. Relat. Fields 4, 203–272 (1998)
Ney P., Spitzer F.: The martin boundary for random walk. Trans. Am. Math. Soc. 121, 116–132 (1966)
Rockafellar, TR.: Convex analysis. Princeton University Press, Princeton (1997) Reprint of the 1970 original, Princeton Paperbacks
Robert P.: Stochastic Metworks and Queues. Springer, Berlin (2003)
Seneta E.: Nonnegative matrices and Markov chains, 2nd edn. Springer, New York (1981)
Shwartz, A., Weiss, A.: Large deviations for performance analysis. Stochastic Modeling Series. Chapman & Hall, London (1995)
Woess W.: Random Walks on Infinite Graphs and Groups. Cambridge University Press, Cambridge (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ignatiouk-Robert, I. Martin boundary of a reflected random walk on a half-space. Probab. Theory Relat. Fields 148, 197–245 (2010). https://doi.org/10.1007/s00440-009-0228-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0228-4
Keywords
- Martin boundary
- Sample path large deviations
- Random walk
Mathematics Subject Classification (2000)
- Primary 60J50
- Secondary 60F10
- 60J45