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

  • Walter A. Rosenkrantz (a1) and C. C. Y. Dorea (a2)

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'.

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Corresponding author

Postal address: Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, U.S.A.

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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.

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References

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Bellman, R. and Harris, T. (1951) Recurrence times for the Ehrenfest urn model. Pacific J. Math. 1, 179193.
Ethier, S. N. (1978) Differentiability properties of Markov semigroups associated with one-dimensional diffusions. Z. Warscheinlichkeitsth. 45, 225238.
Gihman, I. I. and Skorokhod, A. V. (1972) Stochastic Differential Equations. Springer-Verlag, Berlin.
Ito, K. and Mckean, H. P. (1974) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.
Kac, M. (1947) Random walk and the theory of Brownian motion. Amer. Math. Monthly 54, 369391.
Karlin, S. and Mcgregor, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352376.
Kato, T. (1966) Perturbation Theory for Linear Operators. Springer-Verlag, Berlin.
Kurtz, T. (1969) Extensions of Trotter's operator semigroup approximation theorems. J. Functional Anal. 3, 354375.
Kurtz, T. (1975) Semigroups of conditional shifts and approximation of Markov processes. Ann. Prob. 3, 618642.
Mandl, P. (1968) Analytical Treatment of One-dimensional Markov Processes. Springer-Verlag, Berlin.
Mcneil, D. R. and Schach, S. (1973) Central limit analogues for Markov population processes. J. R. Statist. Soc. B 35, 123.
Rosenkrantz, W. (1974) A convergent family of diffusion processes whose diffusion coefficients diverge. Bull. Amer. Math. Soc. 80, 973976.
Rosenkrantz, W. (1975) Limit theorems for solutions to a class of stochastic differential equations. Indiana Math. J. 24, 613625.
Rosenkrantz, W. (1977) Lectures on Markov processes and their associated semigroups. Mimeo series No. 488, Department of Statistics, Purdue University.
Skorokhod, A. V. (1958) Limit theorems for Markov processes. Theory Prob. Appl. 3, 202246.
Stone, C. (1961) Limit Theorems for Birth and Death Processes and Diffusion Processes. Ph.D. Thesis, Department of Mathematics, Stanford University.
Trotter, H. (1958) Approximation of semigroups of operators. Pacific J. Math. 8, 887919.

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

  • Walter A. Rosenkrantz (a1) and C. C. Y. Dorea (a2)

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