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The Ladder Variables of a Markov Random Walk Gerold Alsmeyer
 

Summary: 1
The Ladder Variables of a Markov Random Walk
Gerold Alsmeyer
Institut f¨ur Mathematische Statistik
Fachbereich Mathematik
Westf¨alische Wilhelms-Universit¨at M¨unster
Einsteinstraße 62
D-48149 M¨unster, Germany
Given a Harris chain (Mn)n0 on any state space (S, S) with essentially
unique stationary measure , let (Xn)n0 be a sequence of real-valued
random variables which are conditionally independent, given (Mn)n0,
and satisfy P(Xk ·|(Mn)n0) = Q(Mk-1, Mk, ·) for some stochastic
kernel Q : S2 × B [0, 1] and all k 1. Denote by Sn the n-th par-
tial sum of this sequence. Then (Mn, Sn)n0 forms a so-called Markov
random walk with driving chain (Mn)n0. Its stationary mean drift is
given by µ = EX1 and assumed to be positive in which case the as-
sociated (strictly ascending) ladder epochs 0 = inf{k 0 : Sk 0},
n = inf{k > n-1 : Sk > Sn-1 } for n 1, and the ladder heights
S
n = Sn for n 0 are a.s. positive and finite random variables. Put

  

Source: Alsmeyer, Gerold - Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster

 

Collections: Mathematics