Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T01:03:14.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit behaviour for stochastic monotonicity and applications

Published online by Cambridge University Press:  01 July 2016

Harry Cohn
Harry Cohn*
Affiliation:
The University of Melbourne
*
∗Postal address: Department of Statistics, University of Melbourne, Parkville, VIC 3052, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

A transition probability kernel P(·,·) is said to be stochastically monotone if P(x, (–∞, y]) is non-increasing in x for every fixed y. A Markov chain is said to be stochastically monotone (SMMC) if its transition probability kernels are stochastically monotone. A new method for tackling the asymptotics of SMMC is given in terms of some limit variables {Wq}. In the temporally homogeneous case a cyclic pattern for {Wq} will describe the limit behaviour of suitably normed and centred processes. As a consequence, geometrically growing constants turn out to pertain to almost sure convergence. Some convergence criteria are given and applications to branching processes and diffusions are outlined.

Keywords

MARKOV CHAINMARTINGALEDIFFUSIONBRANCHING
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 331 - 347
Copyright
Copyright © Applied Probability Trust 1988 

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.)

Footnotes

Research partly carried out while visiting the Center for Stochastic Processes, University of North Carolina and supported by AFOSR # F49620 82 C 0009.

References

[1] Aldous, D. (1983) Tail behaviour of birth-and-death and stochastically monotone sequences. Z. Wahrscheinlichkeitsth. 62, 375–394.CrossRefGoogle Scholar
[2] Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhauser, Boston.CrossRefGoogle Scholar
[3] Cohn, H. (1977) On the norming constants occurring in convergent Markov chains. Bull. Austral. Math. Soc. 17, 193–205.CrossRefGoogle Scholar
[4] Cohn, H. (1981) On the convergence of stochastically monotone sequences of random variables and applications. J. Appl. Prob. 18, 592–605.CrossRefGoogle Scholar
[5] Cohn, H. (1982) On a property related to convergence in probability and some applications to branching processes. Stoch. Proc. Appl. 12, 58–72.Google Scholar
[6] Cohn, H. (1979) On the invariant events of a Markov chain. Z. Wahrscheinlichkeitsth. 48, 81–96.CrossRefGoogle Scholar
[7] Cohn, H. (1983) On the fluctuation of stochastically monotone Markov chains and some applications. J. Appl. Prob. 20, 178–184.CrossRefGoogle Scholar
[8] Cohn, H. (1986) Limit behaviour for stochastic monotonicity and applications. Melbourne University, Department of Statistics Research Report Series 8.Google Scholar
[9] Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
[10] Daley, D. J. (1968) Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.CrossRefGoogle Scholar
[11] Gikhman, I. I. and Skorohod, A. V. (1972) Stochastic Differential Equations. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[12] Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
[13] Kamae, T. and Krengel, U. (1978) Stochastic partial ordering. Ann. Prob. 6, 1044–1049.CrossRefGoogle Scholar
[14] Kamae, T., Krengel, U. and O&Brien, G. L. (1977) Stochastic inequalities on partially ordered sets. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
[15] Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
[16] Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.CrossRefGoogle Scholar
[17] Kingman, J. F. C. (1963) Ergodic properties of continuous time Markov processes and their discrete skeletons. Proc. London Math. Soc. 13, 593–604.CrossRefGoogle Scholar
[18] Klebaner, F. C. (1984) On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.CrossRefGoogle Scholar
[19] Kuster, P. (1983) Generalized Markov branching proceses with state dependent offspring distribution. Z. Wahrscheinlichkeitsth. 64, 475–503.CrossRefGoogle Scholar
[20] Lindvall, T. (1974) Limit theorems for some functionals of certain Galton-Watson processes. Adv. Appl. Prob. 2, 309–321.CrossRefGoogle Scholar
[21] Loève, M. (1977) Probability Theory I, 4th edn. Springer-Verlag, New York.Google Scholar
[22] Rényi, A. (1958) On mixing sequences of sets. Acta Math. Acad. Hungar. 9, 215–228.CrossRefGoogle Scholar
[23] Rosler, U. (1982) The Martin Boundary for Time-Dependent Omstein-Uhlenbeck Processes. Habilitation Thesis, Gottingen.Google Scholar
[24] Schuh, H.-J. and Barbour, A. (1977) On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681–723.CrossRefGoogle Scholar
[25] Stone, C. (1963) Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7, 638–660.CrossRefGoogle Scholar
[26] Stoyan, D. (1984) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar