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December, 1974 A Central Limit theorem for Markov Processes that Move by Small Steps
M. Frank Norman
Ann. Probab. 2(6): 1065-1074 (December, 1974). DOI: 10.1214/aop/1176996498


We consider a family $X_n^\theta$ of discrete-time Markov processes indexed by a positive "step-size" parameter $\theta$. The conditional expectations of $\Delta X_n^\theta, (\Delta X_n^\theta)^2$, and $|\Delta X_n^\theta|^3$, given $X_n^\theta$, are of the order of magnitude of $\theta, \theta^2$, and $\theta^3$, respectively. Previous work has shown that there are functions $f$ and $g$ such that $(X_n^\theta - f(n\theta))/\theta^{\frac{1}{2}}$ is asymptotically normally distributed, with mean 0 and variance $g(t)$, as $\theta \rightarrow 0$ and $n\theta \rightarrow t < \infty$. The present paper extends this result to $t = \infty$. The theory is illustrated by an application to the Wright-Fisher model for changes in gene frequency.


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M. Frank Norman. "A Central Limit theorem for Markov Processes that Move by Small Steps." Ann. Probab. 2 (6) 1065 - 1074, December, 1974.


Published: December, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0294.60054
MathSciNet: MR368150
Digital Object Identifier: 10.1214/aop/1176996498

Primary: 60F05
Secondary: 60J05 , 92A10

Keywords: central limit theorem , Markov process , step-size parameter

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 6 • December, 1974
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