Recurrence theorems for Markov random walks

• Autorzy: Alsmeyer G.
• Języki publikacji: EN

Abstrakty

EN

Let (M,_n, S_n)_n≥0 be a Markov random walk whose driving chain (M_n)_n≥0 with general state space (ℒ,Ϭ) is ergodic with unique stationary distribution ξ. Providing n⁻¹ S_n→o in probability under P_ξ, it is shown that the recurrence set of (S_n−γ(M_o) +γ(M_n))_n≥o forms a closed subgroup of Rdepending on the lattice-type of (M_n, Sn)_n≥o. The so-called shift function γ is bounded and appears in that lattice-type condition. The recurrence set of (S_n)_n≥o itself is also given but may lookmore complicated depending on γ. The results extend the classical recurrence the orem for random walks with i.i.d. increments and further sharpenresults by Berbee, Dekking and others on the recurrence behavior of random walks with stationary increments.

Słowa kluczowe

EN

Markov random walk; random walk; stationary increments; recurrence point

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2001
• Tom: Vol. 21, Fasc. 1
• Strony: 123--134
• Opis fizyczny: Bibliogr.11 poz.

Twórcy

autor

Alsmeyer G.

Bibliografia

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• [9] V. M. Shurenkov, On the theory of Markov renewal, Theory Probab. Appl. 29 (1984), pp. 247-265.
• [10] C. J. Stone, On the potential operator for one-dimensional recurrent random walks, Trans. Amer. Math. Soc. 136 (1969), pp. 427-445.
• [11] H. Thorisson, Coupling, Stationarity, and Regeneration, Springer, New York 2000.