Random walks in a quarter plane with zero drifts. I. Ergodicity and null recurrence
Annales de l'I.H.P. Probabilités et statistiques, Volume 28 (1992) no. 2, p. 179-194
@article{AIHPB_1992__28_2_179_0,
author = {Fayolle, G. and Malyshev, V. A. and Menshikov, M. V.},
title = {Random walks in a quarter plane with zero drifts. I. Ergodicity and null recurrence},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {28},
number = {2},
year = {1992},
pages = {179-194},
zbl = {0747.60064},
mrnumber = {1162572},
language = {en},
url = {http://www.numdam.org/item/AIHPB_1992__28_2_179_0}
}

Fayolle, G.; Malyshev, V. A.; Menshikov, M. V. Random walks in a quarter plane with zero drifts. I. Ergodicity and null recurrence. Annales de l'I.H.P. Probabilités et statistiques, Volume 28 (1992) no. 2, pp. 179-194. http://www.numdam.org/item/AIHPB_1992__28_2_179_0/

[1] V.A. Malyshev, Classification of Two-Dimensional Positive Random Walks and Almost Linear Semimartingales, Dokl. Akad. Nauk. S.S.S.R., Vol. 202, 1972, pp. 526- 528 ; English Transl., Soviet Math. Dokl., Vol. 13, 1972. | MR 309202 | Zbl 0246.60055

[2] V.A. Malyshev and M.V. Menshikov, Ergodicity, Continuity and Analyticity of Countable Markov Chains, Trans. Moscow Math. Soc., Vol. 39, 1979, pp. 2-48; English Transl., 1981, Issue I. | MR 544940 | Zbl 0459.60059

[3] K.L. Vaninsky and B.V. Lazareva, About Ergodicity and Transience of Homogeneous Markov Chains in a Positive Quadrant, Prob. Inf. Trans., Vol. 24, No. 1, 1988, pp. 105-110. | MR 939579 | Zbl 0676.60066

[4] G. Fayolle and R. Iasnogorodski, Criteria for the Non-Ergodicity of Stochastic Processes: Application to the Exponential Back-off Protocol, J. Appl. Prob., Vol. 24, (1987), pp. 347-354. | MR 889799 | Zbl 0625.60076

[5] G. Fayolle, On Random Walks Arising in Queueing Systems: Ergodicity and Transience via Quadratic Forms as Lyapounov Functions, Part I, Queueing Systems Theory Appl., Vol. 5, 1989, pp. 167-184. | MR 1032553 | Zbl 0687.60083

[6] M.V. Menshikov, Ergodicity and Transience Conditions for Random Walks in the Positive Octant of Space, Dokl. Akad. Nauk S.S.S.R., Vol. 217, 1974, pp. 755-758, English Transl., Soviet Math. Dokl., Vol. 15, 1974. | MR 359005 | Zbl 0309.60042

[7] S. Karlin, A First Course in Stochastic Processes, Academic Press, 1966. | MR 208657 | Zbl 0177.21102

[8] V.A. Malyshev, Wiener-Hopf equations, Galois Automorphisms, Random Walks in a Quarter Plane, Moscow State Univ., 1970. | MR 428464

[9] V.A. Malyshev, Doctor Thesis, Moscow, 1973.

[10] F.G. Foster, On Stochastic Matrices Associated with Certain Queueing Processes, Ann. Math. Statist., Vol. 24, 1953, pp. 355-360. | MR 56232 | Zbl 0051.10601

[11] W.A. Rosenkrantz, Ergodicity Conditions for Two-Dimensional Markov Chains on the Positive Quadrant, Prob. Theory Related Fields, Vol. 83, No. 3, 1989. | MR 1017399 | Zbl 0661.60083

[12] R.L. Tweedie, Criteria for Classifying General Markov Chains, Adv. Appl. Prob., Vol. 8, 1976, pp. 737-771. | MR 451409 | Zbl 0361.60014

[13] M. Loeve, Probability Theory, 2nd. Edition, Princeton, van Nostrand, 1960. | MR 123342 | Zbl 0095.12201

[14] L.A. Klein Haneveld and A.O. Pittinger, Escape Time for a Random Walk from an Orthant, Stochastic Processes and their Applications, Vol. 35, 1990, pp. 1-9. | MR 1062579 | Zbl 0702.60062

[15] S.R.S. Varadhan and R.J. Williams, Brownian Motion in a Wedge with Oblique Reflection, Comm. Pure Appl. Math., Vol. 38, 1985, pp. 405-443. | MR 792398 | Zbl 0579.60082

[16] R.J. Williams, Recurrence Classification and Invariant Measure for Reflected Brownian Motion in a Wedge, Annals Prob., Vol. 13, No. 3, 1985, pp. 758-778. | MR 799421 | Zbl 0596.60078