A central limit theorem for Markov chains and applications to hypergroups

Abstract

Let ( X n ) (X_n) be a homogeneous Markov chain on an unbounded Borel subset of R \mathbb {R} with a drift function d d which tends to a limit m 1 m_1 at infinity. Under a very simple hypothesis on the chain we prove that n − 1 / 2 ( X n − ∑ k = 1 n d ( X k − 1 ) ) \displaystyle n^{-1/2} (X_n - \sum ^ n_{k=1} d(X_{k-1})) converges in distribution to a normal law N ( 0 , σ 2 ) N (0, \sigma ^2) where the variance σ 2 \sigma ^2 depends on the asymptotic behaviour of ( X n ) (X_n) . When d − m 1 d - m_1 goes to zero quickly enough and m 1 ≠ 0 m_1 \neq 0 , the random centering may be replaced by n m 1 . n m_1. These results are applied to the case of random walks on some hypergroups.