Ladder epochs and ladder chain of a Markov random walk with discrete driving chain
| Autor: | Gerold Alsmeyer |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
Discrete mathematics Stationary distribution Markov chain Applied Mathematics Probability (math.PR) 010102 general mathematics Coupling (probability) Random walk 60J10 60K15 01 natural sciences 010104 statistics & probability Chain (algebraic topology) Factorization FOS: Mathematics State space Almost surely 0101 mathematics Mathematics - Probability Mathematics |
| Zdroj: | Advances in Applied Probability. 50:31-46 |
| ISSN: | 1475-6064 0001-8678 |
| Popis: | Let (Mn,Sn)n≥0 be a Markov random walk with positive recurrent driving chain (Mn)n≥0 on the countable state space 𝒮 with stationary distribution π. Suppose also that lim supn→∞Sn=∞ almost surely, so that the walk has almost-sure finite strictly ascending ladder epochs σn>. Recurrence properties of the ladder chain (Mσn>)n≥0 and a closely related excursion chain are studied. We give a necessary and sufficient condition for the recurrence of (Mσn>)n≥0 and further show that this chain is positive recurrent with stationary distribution π> and 𝔼π>σ1>n,𝑆̂n)n≥0, obtained by time reversal and called the dual of (Mn,Sn)n≥0, is positive divergent, i.e. 𝑆̂n→∞ almost surely. Simple expressions for π> are also provided. Our arguments make use of coupling, Palm duality theory, and Wiener‒Hopf factorization for Markov random walks with discrete driving chain. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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