Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T11:48:40.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for stationary distributions

Published online by Cambridge University Press:  01 July 2016

M. Frank Norman
M. Frank Norman*
Affiliation:
University of Pennsylvania
Get access Rights & Permissions [Opens in a new window]

Abstract

For every N ≧ 1, let be a real-valued stationary process, and let . Suppose that and var, where ∊N and τN are positive null sequences. Limiting distributions of as N → ∞ are obtained for the cases τN = ∊N and τN = o(∊N). These results are established by an extension of a method due to Moran. The theory is illustrated by a variety of applications to genetic models.

Keywords

STATIONARY PROCESSESLIMIT THEOREMSDIFFUSION APPROXIMATIONGENETIC MODELS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 3 , September 1975 , pp. 561 - 575
Copyright
Copyright © Applied Probability Trust 1975 

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

Feller, W. (1951) Diffusion processes in genetics. Proc. Second Berkeley Symp. Math. Statist. Prob. 227–246. University of California Press, Berkeley.Google Scholar
Karlin, S., and McGregor, J. (1964) On some stochastic models in genetics. Stochastic Models in Medicine and Biology, ed. Gurland, J., pp. 245–271. University of Wisconsin Press, Madison.Google Scholar
Moran, P. A. P. (1958a) A general theory of the distribution of gene frequencies I. Overlapping generations. Proc. Roy. Soc. London B 149, 102–112.Google Scholar
Moran, P. A. P. (1958b) A general theory of the distribution of gene frequencies II. Non-overlapping generations. Proc. Roy. Soc. London B 149, 113–116.Google Scholar
Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
Norman, M. F. (1972) Markov Processes and Learning Models. Academic Press, New York.Google Scholar
Norman, M. F. (1974) A central limit theorem for Markov processes that move by small steps. Ann. Probability 2, 1065–1074.Google Scholar
Norman, M. F. (1975a) Diffusion approximation of non-Markovian processes. Ann. Probability 3, 358–364.CrossRefGoogle Scholar
Norman, M. F. (1975b) Approximation of stochastic processes by Gaussian diffusions, and applications to Wright-Fisher genetic models. SIAM J. Appl. Math. 29, 225–242.Google Scholar
Wright, S. (1938) The distribution of gene frequencies in populations of polyploids. Proc. Nat. Acad. Sci., Wash. 24, 372–377.Google Scholar