Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T02:48:09.654Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotonicity of Positive Dependence with Time for Stationary Reversible Markov Chains

Published online by Cambridge University Press:  27 July 2009

Taizhong Hu  and
Harry Joe
Taizhong Hu
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
Harry Joe
Affiliation:
Department of Statistics, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z2
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (X1, X2) and (Y1, Y2) be bivariate random vectors with a common marginal distribution (X1, X2) is said to be more positively dependent than (Y1, Y2) if E[h(X1)h(X2)] ≥ E[h(Y1)h(Y2)] for all functions h for which the expectations exist. The purpose of this paper is to study the monotonicity of positive dependence with time for a stationary reversible Markov chain [X1]; that is, (Xs, Xl+s) is less positively dependent as t increases. Both discrete and continuous time and both a denumerable set and a subset of the real line for the state space are considered. Some examples are given to show that the assertions established for reversible Markov chains are not true for nonreversible chains.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 2 , April 1995 , pp. 227 - 237
Copyright
Copyright © Cambridge University Press 1995

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.Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring, MD: To Begin With. (First printed in 1975.)Google Scholar
2.Bhattacharya, R.N. & Waymire, E.C. (1990). Stochastic processes with applications. New York: John Wiley and Sons.Google Scholar
3.Fang, Z.B., Hu, T.Z., & Joe, H. (1994). On the decrease in dependence with lag for stationary Markov chains. Probability in the Engineering and Informational Sciences 8: 385–401.CrossRefGoogle Scholar
4.Gleser, L.J. & Moore, D.S. (1985). Positive dependence in Markov chains. Linear Algebra Applications 70: 131–146.CrossRefGoogle Scholar
5.Joe, H. (1993). Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis 46: 262–282.CrossRefGoogle Scholar
6.Marcus, M. & Mine, H. (1964). A survey of matrix theory and matrix inequalities. Boston: Prindle, Weber and Schmidt.Google Scholar
7.Rinott, Y. & Pollak, M. (1980). A strong ordering induced by a concept of positive dependence and monotonicity of asymptotic test sizes. Annals of Statistics 8: 190–198.CrossRefGoogle Scholar
8.Ross, S.M. (1983). Stochastic processes. New York: Wiley.Google Scholar
9.Shaked, M. (1979). Some concepts of positive dependence for bivariate interchangeable distributions. Annals of the Institute of Statistical Mathematics 31: 67–84.CrossRefGoogle Scholar
10.Tchen, A.H. (1980). Inequalities for distributions with given marginals. Annals of Probability 8: 814–827.CrossRefGoogle Scholar