Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T09:31:07.362Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for Markov processes via a variant of the Trotter-Kato theorem

Published online by Cambridge University Press:  14 July 2016

Walter A. Rosenkrantz  and
C. C. Y. Dorea
Walter A. Rosenkrantz*
Affiliation:
University of Massachusetts
C. C. Y. Dorea
Affiliation:
Universidade de Brasilia
*
∗Postal address: Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

A variant of the Trotter–Kato theorem due to Kurtz (1969) is used to give new and simpler proofs of functional central limit theorems for Markov processes. Applications include theorems of Bellman and Harris (1951), Stone (1961), Karlin and McGregor (1965), Gihman and Skorokhod (1972) and Rosenkrantz (1975). In addition our methods yield a novel counterexample to the so-called ‘diffusion approximation'.

Keywords

MARKOV PROCESSESFUNCTIONAL CENTRAL LIMIT THEOREMSTROTTER-KATO THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 704 - 715
Copyright
Copyright © Applied Probability Trust 1980 

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 supported by U.S. Air Force Office of Scientific Research under Contract F4962079-C-0209.

∗∗

Universidade de Brasilia, Departamento de Mathematica, 1E 70.000 Brasilia DF, Brazil.

References

Bellman, R. and Harris, T. (1951) Recurrence times for the Ehrenfest urn model. Pacific J. Math. 1, 179–193.CrossRefGoogle Scholar
Ethier, S. N. (1978) Differentiability properties of Markov semigroups associated with one-dimensional diffusions. Z. Warscheinlichkeitsth. 45, 225–238.CrossRefGoogle Scholar
Gihman, I. I. and Skorokhod, A. V. (1972) Stochastic Differential Equations. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Ito, K. and Mckean, H. P. (1974) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.Google Scholar
Kac, M. (1947) Random walk and the theory of Brownian motion. Amer. Math. Monthly 54, 369–391.CrossRefGoogle Scholar
Karlin, S. and Mcgregor, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352–376.CrossRefGoogle Scholar
Kato, T. (1966) Perturbation Theory for Linear Operators. Springer-Verlag, Berlin.Google Scholar
Kurtz, T. (1969) Extensions of Trotter's operator semigroup approximation theorems. J. Functional Anal. 3, 354–375.CrossRefGoogle Scholar
Kurtz, T. (1975) Semigroups of conditional shifts and approximation of Markov processes. Ann. Prob. 3, 618–642.CrossRefGoogle Scholar
Mandl, P. (1968) Analytical Treatment of One-dimensional Markov Processes. Springer-Verlag, Berlin.Google Scholar
Mcneil, D. R. and Schach, S. (1973) Central limit analogues for Markov population processes. J. R. Statist. Soc. B 35, 1–23.Google Scholar
Rosenkrantz, W. (1974) A convergent family of diffusion processes whose diffusion coefficients diverge. Bull. Amer. Math. Soc. 80, 973–976.CrossRefGoogle Scholar
Rosenkrantz, W. (1975) Limit theorems for solutions to a class of stochastic differential equations. Indiana Math. J. 24, 613–625.CrossRefGoogle Scholar
Rosenkrantz, W. (1977) Lectures on Markov processes and their associated semigroups. Mimeo series No. 488, Department of Statistics, Purdue University.Google Scholar
Skorokhod, A. V. (1958) Limit theorems for Markov processes. Theory Prob. Appl. 3, 202–246.CrossRefGoogle Scholar
Stone, C. (1961) Limit Theorems for Birth and Death Processes and Diffusion Processes. Ph.D. Thesis, Department of Mathematics, Stanford University.Google Scholar
Trotter, H. (1958) Approximation of semigroups of operators. Pacific J. Math. 8, 887–919.CrossRefGoogle Scholar