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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A central limit theorem for Markov chains
and applications to hypergroups


Author: Léonard Gallardo
Journal: Proc. Amer. Math. Soc. 127 (1999), 1837-1845
MSC (1991): Primary 60J10, 60F05, 60J15
Published electronically: February 23, 1999
MathSciNet review: 1476127
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Abstract: Let $(X_n)$ be a homogeneous Markov chain on an unbounded Borel subset of $\mathbb{R}$ with a drift function $d$ which tends to a limit $m_1$ at infinity. Under a very simple hypothesis on the chain we prove that $\displaystyle n^{-1/2} (X_n - \sum^ n_{k=1} d(X_{k-1}))$ converges in distribution to a normal law $N (0, \sigma^2)$ where the variance $\sigma^2$ depends on the asymptotic behaviour of $(X_n)$. When $d - m_1$ goes to zero quickly enough and $m_1 \neq 0$, the random centering may be replaced by $n m_1.$ These results are applied to the case of random walks on some hypergroups.


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Additional Information

Léonard Gallardo
Affiliation: Departement de Mathematiques, Université de Tours, Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France
Email: gallardo@univ-tours.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04665-1
PII: S 0002-9939(99)04665-1
Received by editor(s): April 14, 1997
Received by editor(s) in revised form: September 22, 1997
Published electronically: February 23, 1999
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1999 American Mathematical Society