Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 1 ,pp. 89--100

PALEY-TYPE INEQUALITIES RELATED TO THE CENTRAL LIMIT THEOREM FOR MARKOV CHAINS

By

C.D. FUH Academia Sinica, Taipei

SUMMARY. Let {(Xn,Sn),n >=0} be a Markov additive process on a general state space $(S, \cal{S})$, with transition kernel P(x,A * B) and invariant probability p for {Xn}. Under some regularity conditions, the central limit theorem holds for $\frac{1}{\sigma \sqrt{n}} (S_n - n E_\pi S_0)$, where Ep refers to the expectation of {Xn} under the initial state X0 has distribution p(.), and s2 is the asymptotic variance of Sn$/\sqrt{n}$. In this paper, we derive a one-sided Paley-type inequality related to the error estimate $\Delta_n = \sup_z |P_\pi( \frac{1}{\sigma \sqrt{n}} (S_n - n E_\pi S_0) \leq z ) - \Phi(z) |$, where Pp refers to the probability of {Xn} under the initial state X0 has distribution p(.), and F(z) is the standard normal distribution.

AMS (1991) subject classification.60F05, 60G50.

Key words and phrases. Berry-Esseen theorem, central limit theorem, Markov chains, Paley inequality, rate of convergence.

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