Renewal theorem for a class of stationary sequences
type: Item
instance of: scholarly article
title: Renewal theorem for a class of stationary sequences
zbMATH Open document ID: 0597.60083
DOI: 10.1007/BF00699103
publication date: 1986-01-01T00:00:00Z
review text: Let \(\xi_ n\), \(n\in {\mathbb{Z}}\), be a sequence of stationary r.v. This sequence is assumed to be of the special structure \(\xi_ n=\xi (T^ n(X))\), T the shift operator and \(X=(...,X_{-1},X_ 0,X_ 1,...)\) a sequence of i.i.d. r.v. The main result is an asymptotic renewal theorem for the overshoot \(S_{\tau}-b\) of the partial sums \(S_ n=\sum^{n}_{i=1}\xi_ i\), \(\tau\) the first time n with \(S_ n>b\). The overshoot \(S_{\tau}-b\) and \(X_ 0\) are asymptotically independent as \(b\to \infty.\) This result on stationary sequences includes known results for recurrent Markov chains and processes, functionals of stationary Gaussian processes, and functionals of one-dimensional Gibbs states. The main argument in the proof is a nice coupling argument for the process X.
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 3962919
zbMATH Keywords: overshot distribution
zbMATH Keywords: asymptotic renewal theorem
zbMATH Keywords: functionals of stationary Gaussian processes
zbMATH Keywords: functionals of one-dimensional Gibbs states
zbMATH Keywords: coupling argument
author: Steven P. Lalley
reviewed by: Uwe Rösler
MaRDI profile type: Publication
cites work: Q5805270
cites work: Limit theorems for semi-Markov processes and renewal theory for Markov chains
cites work: A New Approach to the Limit Theory of Recurrent Markov Chains
cites work: Q3886625
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Q3955608
cites work: Q5621834
cites work: Ergodic automorphisms of \(T^ n\) are Bernoulli shifts
cites work: Renewal theory for functionals of a Markov chain with general state space
cites work: Conditional Markov renewal theory. I. Finite and denumerable state space
cites work: Regenerative representation for one-dimensional Gibbs states
cites work: A probabilistic proof of Blackwell's renewal theorem
cites work: A refinement of the coupling method in renewal theory
cites work: Change of Time Scale For Markov Processes
cites work: Q4744504
cites work: Bernoulli flows over maps of the interval
Recommended article: Q3761439
Recommended article: Some Renewal Type Theorems
Recommended article: A renewal theorem in the finite-mean case
Recommended article: Markov renewal theorems in a scheme of series
Recommended article: Renewal theorem for a system of renewal equations
Recommended article: A renewal theorem in the infinite mean case
Recommended article: Q4940653
Recommended article: Q3319543
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3962919