Finite-time implications of relaxation times for stochastically monotone processes
type: Item
instance of: scholarly article
title: Finite-time implications of relaxation times for stochastically monotone processes
zbMATH Open document ID: 0644.60070
DOI: 10.1007/BF01848136
author: David Aldous
publication date: 1988-01-01T00:00:00Z
review text: A continuous-time Markov chain \((X_ t)\) which converges to a stationary distribution \(\pi\) often does so asymptotically at an exponential rate, that is, \(P(X_ t\in A)-\pi (A)\sim c_ A\exp (-\lambda t)\) as \(t\to \infty\). The quantity \(\tau\equiv 1/\lambda\) is then called the relaxation time of the chain. Motivated by the general question of what can be said rigorously about the finite-time behaviour of the chain given its stationary distribution \(\pi\) and its relaxation time \(\tau\), the author obtains a result for stochastically monotone chains for which the reversed chain is also stochastically monotone to the effect that \[ \sum_{i}\pi_ i\max_{j}| P_ i(X_ t\leq j)-\pi [0,j]| \leq 2(2+t/\tau)\exp (-t/t). \] Stochastic monotonicity is used here in the sense of monotonicity ``in space''.
Mathematics Subject Classification ID: 60J27
zbMATH DE Number: 4050685
zbMATH Keywords: relaxation time
zbMATH Keywords: stationary distribution
zbMATH Keywords: stochastically monotone chains
zbMATH Keywords: Stochastic monotonicity
reviewed by: Erik A. van Doorn
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4050685