Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T02:11:01.862Z Has data issue: false hasContentIssue false
  • English
  • Français

The eigenvalues of the empirical transition matrix of a Markov chain

Part of: Limit theorems Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Geoffrey Pritchard  and
David J. Scott
Geoffrey Pritchard
Affiliation:
Department of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand. Email address: [email protected]
David J. Scott
Affiliation:
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

This paper investigates the probabilistic behaviour of the eigenvalue of the empirical transition matrix of a Markov chain which is of largest modulus other than 1, loosely called the second-largest eigenvalue. A central limit theorem is obtained for nonmultiple eigenvalues of the empirical transition matrix. When the Markov chain is actually a sequence of independent observations the distribution of the second-largest eigenvalue is determined and a test for independence is developed. The independence case is considered in more detail when the Markov chain has only two states, and some applications are given.

Keywords

Markov chaineigenvaluerandom matrix

MSC classification

Primary: 62M05: Markov processes
Secondary: 60F05: Central limit and other weak theorems 62M02: Markov processes
Type
Part 6. Stochastic processes
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 347 - 360
Copyright
Copyright © Applied Probability Trust 2004 

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

[1] Antipov, M. V., Izraῐlev, F. M. and Čirikov, B. V. (1968). Statistical testing of a pseudo-random number generator. Vyčisl. Sistemy 30, 77–85 (in Russian).Google Scholar
[2] Asmussen, S., Glynn, R W. and Thorisson, H. (1992). Stationarity detection in the initial transient problem. ACM Trans. Modelling Comput. Simul. 2, 130–157.Google Scholar
[3] Bartlett, M. S. (1951). The frequency goodness of fit test for probability chains. Proc. Camb. Phil. Soc. 47, 86–95.Google Scholar
[4] Billingsley, P. (1961). Statistical methods in Markov chains. Ann. Math. Statist. 32, 12–40.Google Scholar
[5] Billingsley, P. (1995). Probability and Measure , 3rd edn. John Wiley, New York.Google Scholar
[6] Cohen, J., Kesten, H. and Newman, C., (eds) (1986). Random Matrices and Their Applications (Contemporary Math. 50), American Mathematical Society, Providence, RI.Google Scholar
[7] Edelman, A. (1997). The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60, 203–232.CrossRefGoogle Scholar
[8] Edelman, A., Kostlan, E, and Shub, M. (1994). How many eigenvalues of a random matrix are real? J. Amer. Math. Soc. 7, 247–267.Google Scholar
[9] Garren, S. T. and Smith, R. L. (2002). Estimating the second largest eigenvalue of a Markov transition matrix. Bernoulli 6, 215–242.Google Scholar
[10] Good, I. J. (1953). The serial test for sampling numbers and other tests for randomness. Proc. Camb. Phil. Soc. 49, 276–284.CrossRefGoogle Scholar
[11] Good, I. J. (1957). On the serial test for random sequences. Ann. Math. Statist. 28, 262–264.Google Scholar
[12] Kato, T., Wu, L. and Yanagihara, N. (1996) The serial test for a nonlinear pseudorandom number generator. Math. Comp. 65, 761–769.Google Scholar
[13] Kendall, M. G. and Smith, B. B. (1938). Randomness and random sampling numbers. J. R. Statist. Soc. 101, 147–166.Google Scholar
[14] Lehmann, N. and Sommers, H. (1991). Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67, 941–944.Google Scholar
[15] Lovász, L. and Winkler, P. (1995). Exact mixing in an unknown Markov chain. Electron. J. Combinatorics 2, No. R15. Available at http://www.combinatorics.org/.Google Scholar
[16] Niederreiter, H. (1978). The serial test for linear-congruential pseudo-random numbers. Bull. Amer. Math. Soc. 84, 273–274.CrossRefGoogle Scholar
[17] Parzen, E. (1964). Stochastic Processes. Holden-Day, San Francisco, CA.Google Scholar
[18] Propp, J. G. and Wilson, D. B. (1998). How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms 27, 170–217.CrossRefGoogle Scholar
[19] Scott, D. J., Lai, C. D. and Wang, D. Q. (1997). All nonunit eigenvalues in a Markov chain being zero does not imply independence. Preprint, Department of Statistics, University of Auckland.Google Scholar
[20] Seneta, E. (1981). Non-Negative Matrices and Markov Chains , 2nd edn. Springer, New York.Google Scholar
[21] Silverstein, J. (1995). Strong convergence of the empirical distribution of eigenvalues of large random matrices. J. Multivariate Anal. 55, 331–339.CrossRefGoogle Scholar
[22] Silverstein, J. and Bai, Z. (1995). On the empirical distribution of eigenvalues of a class of large-dimensional random matrices. J. Multivariate Anal. 54, 175–192.Google Scholar
[23] Sommers, H. J., Crisanti, A., Sompolinsky, H. and Stein, Y. (1988). Spectrum of large random asymmetric matrices. Phys. Rev. Lett. 60, 1895–1898.Google Scholar
[24] Wang, D. Q. and Scott, D. J. (1989). Testing a Markov chain for independence. Commun. Statist. Theory Meth. 18, 4085–4103.CrossRefGoogle Scholar
[25] Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem. Clarendon Press, Oxford.Google Scholar