Random walks in a quarter plane with zero drifts. I. Ergodicity and null recurrence

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1. [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. Zbl0246.60055MR309202
2. [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. Zbl0459.60059MR544940
3. [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. Zbl0676.60066MR939579
4. [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. Zbl0625.60076MR889799
5. [5] G. Fayolle, On RandomWalks 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. Zbl0687.60083MR1032553
6. [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. Zbl0309.60042MR359005
7. [7] S. Karlin, A First Course in Stochastic Processes, Academic Press, 1966. Zbl0177.21102MR208657
8. [8] V.A. Malyshev, Wiener-Hopf equations, Galois Automorphisms, Random Walks in a Quarter Plane, Moscow State Univ., 1970. MR428464
9. [9] V.A. Malyshev, Doctor Thesis, Moscow, 1973. 
10. [10] F.G. Foster, On Stochastic Matrices Associated with Certain Queueing Processes, Ann. Math. Statist., Vol. 24, 1953, pp. 355-360. Zbl0051.10601MR56232
11. [11] W.A. Rosenkrantz, Ergodicity Conditions for Two-Dimensional Markov Chains on the Positive Quadrant, Prob. Theory Related Fields, Vol. 83, No. 3, 1989. Zbl0661.60083MR1017399
12. [12] R.L. Tweedie, Criteria for Classifying General Markov Chains, Adv. Appl. Prob., Vol. 8, 1976, pp. 737-771. Zbl0361.60014MR451409
13. [13] M. Loeve, Probability Theory, 2nd. Edition, Princeton, van Nostrand, 1960. Zbl0095.12201MR123342
14. [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. Zbl0702.60062MR1062579
15. [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. Zbl0579.60082MR792398
16. [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. Zbl0596.60078MR799421

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