Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T02:33:57.254Z Has data issue: false hasContentIssue false
  • English
  • Français

Empirical convergence rates for continuous-time Markov chains

Part of: Inference from stochastic processes Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Geoffrey Pritchard  and
David J. Scott
Geoffrey Pritchard*
Affiliation:
University of Auckland
David J. Scott*
Affiliation:
University of Auckland
*
∗ Postal address: Department of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand.
∗∗ Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of estimating the rate of convergence to stationarity of a continuous-time, finite-state Markov chain. This is done via an estimator of the second-largest eigenvalue of the transition matrix, which in turn is based on conventional inference in a parametric model. We obtain a limiting distribution for the eigenvalue estimator. As an example we treat an M/M/c/c queue, and show that the method allows us to estimate the time to stationarity τ within a time comparable to τ.

Keywords

continuous-time Markov chaineigenvalueconvergence rateM/M/c/c queuerandom matrix

MSC classification

Primary: 62M05: Markov processes
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 262 - 269
Copyright
Copyright © by the Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Basawa, I. V., and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic Processes. Academic Press, London.Google Scholar
Billingsley, P. (1961). Statistical Inference for Markov Processes. University of Chicago Press.Google Scholar
Eaton, M. L., and Tyler, D. (1994). The asymptotic distribution of singular values with applications to canonical correlations and correspondence analysis. J. Multivar. Anal. 50, 238–264.CrossRefGoogle Scholar
Garren, S. T., and Smith, R. L. (2000). Estimating the second largest eigenvalue of a Markov transition matrix. Bernoulli 6, 215–242.Google Scholar
Pritchard, G., and Scott, D. J. (2000). The second-largest eigenvalue of an empirical Markov transition matrix. Submitted.Google Scholar
Ruymgaart, F. H., and Yang, S. (1997). Some applications of Watson's perturbation approach to random matrices. J. Multivar. Anal. 60, 48–60.Google Scholar
Wang, D. Q., and Scott, D. J. (1989). Testing a Markov chain for independence. Commun. Statist. Theory Meth. 18, 4085–4103.CrossRefGoogle Scholar