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1

Asymptotic behavior of ultimately contractive iterated random Lipschitz functions

Alsmeyer G., Hölker G.

Probability and Mathematical Statistics

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2009

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Vol. 29, Fasc. 2

321--336

EN

Let (Fn)n≥0 be a random sequence of i.i.d. global Lipschitz functions on a complete separable metric space (X; d) with Lipschit constants L1; L2; : : : For n ≥0, denote by Mx n = Fn○ : : : ○ F1(x) and ^Mx n = Fn○ : : : ○ F1(x) the associated sequences of forward and backward iterations, respectively. If E log+ L1 < 0 (mean contraction) and E log+ d ( F1(x0); x0) is finite for some x0ЄX, then it is known (see [9]) that, for each x Є X, the Markov chain Mx n converges weakly to its unique stationary distribution π, while ^M xn is a.s. convergent to a random variable ^M∞ which does not depend on x and has distribution π. In [2], renewal theoretic methods have been successfully employed to provide convergence rate results for ^M x n, which then also lead to corresponding assertions for Mx n via Mx n d= ^M x n for all n and x, where d= means equality in law. Here our purpose is to demonstrate how these methods are extended to the more general situation where only ultimate contraction, i.e. an a.s. negative Lyapunov exponent limn→∞ n−1 log l(Fn○ : : : ○ F1) is assumed (here l(F) denotes the Lipschitz constant of F). This not only leads to an extension of the results from [2] but in fact also to improvements of the obtained convergence rate.

2

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

Alsmeyer G.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

389--411

EN

Let (Mn)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 Sn (g)def = g (M1) +…+ g (Mn) for n ≥ 1. Given any stopping time T for (Mn)n ≥ 0 and any initial distribution ν for (Mn)n ≥ 0, the purpose of this paper is to provide suitable conditions for the finiteness of Eν ST (g)p for p > 1. A typical result states that Eν ST (g)p ≤ C1 (Eν ST (gp) + Eν Tp) + C2 for suitable finite constants C1, C2. Our analysis is based to a large extent on martingale decompositions for Sn (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 (Mn)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: Sn (g) > t} for t ≥ 0 is also treated.

3

Recurrence theorems for Markov random walks

Alsmeyer G.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 1

123--134

EN

Let (M,n, Sn)n≥0 be a Markov random walk whose driving chain (Mn)n≥0 with general state space (ℒ,Ϭ) is ergodic with unique stationary distribution ξ. Providing n−1 Sn→o in probability under Pξ, it is shown that the recurrence set of (Sn−γ(Mo) +γ(Mn))n≥o forms a closed subgroup of Rdepending on the lattice-type of (Mn, Sn)n≥o. The so-called shift function γ is bounded and appears in that lattice-type condition. The recurrence set of (Sn)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.

4

The ladder variables of a Markov random walk

Alsmeyer G.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 1

151--168

EN

Given a Harris chain (Mn)n≥0 on any state space (S, C) with essentially unique stationary measure ξ, let (Xn)n≥0 be a sequence of real-valued random variables which are conditionally independent, given (Mn)n≥0, and satisfy [formula].. for some stochastic kernel Q : S2 × B → [0, 1] and all k ≥ 1. Denote by Sn the n-th partial sum of this sequence. Then (Mn, Sn)n≥0 forms a so-called Markov random walk with driving chain (Mn, Sn)n≥0. Its stationary mean drift is given by μ = EξX1 and assumed to be positive in which case the associated (strictly ascending) ladder epochs [formula].. and the ladder heights S*n = Sσn for n ≥ 0 are a.s. positive and finite randomvariables. Put M*n = Mσn. ……..[formula]