On the existence of moments of stopped sums in Markov renewal theory

• Autorzy: Alsmeyer G.
• Języki publikacji: EN

Abstrakty

EN

Let (M_n)_{n ≥ 0} be an ergodic Markov chain on a general state space X with stationary distribution π and g: X → [0, ∞) a measurable function. Define S₀ (g)^def = 0 and S_n (g)^def = g (M₁) +…+ g (M_n) for n ≥ 1. Given any stopping time T for (M_n)_{n ≥ 0} and any initial distribution ν for (M_n)_{n ≥ 0}, the purpose of this paper is to provide suitable conditions for the finiteness of E_ν S_T (g)^p for p > 1. A typical result states that E_ν S_T (g)^p ≤ C₁ (E_ν S_T (g^p) + E_ν T^p) + C₂ for suitable finite constants C₁, C₂. Our analysis is based to a large extent on martingale decompositions for S_n (g) and on drift conditions for the function g and the transition kernel P of the chain. Some of the results are stated under the stronger assumption that (M_n)_{n ≥ 0} is positive Harris recurrent in which case stopping times T which are regeneration epochs for the chain are of particular interest. The important special case where T = T(t)^def = inf {n ≥ 1: S_n (g) > t} for t ≥ 0 is also treated.

Słowa kluczowe

EN

Markov random walk; stopped sum; Harris recurrence; regeneration epoch; drift condition; l-dependence; martingale; Burkholder’s inequality

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2003
• Tom: Vol. 23, Fasc. 2
• Strony: 389--411
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Alsmeyer G.

• Institut für Mathematische Statistik, Fachbereich Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstraβe 62, D-48149 Münster, Germany

Bibliografia

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