A central limit theorem for Markov chains and applications to hypergroups

Gallardo, Léonard

doi:10.1090/S0002-9939-99-04665-1

A central limit theorem for Markov chains and applications to hypergroups
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Proc. Amer. Math. Soc. 127 (1999), 1837-1845 Request permission

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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