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 {(X_{n},S_{n}),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 {X_{n}}.
Under some regularity conditions, the central limit theorem
holds for $\frac{1}{\sigma \sqrt{n}} (S_n - n E_\pi S_0)$, where
E_{p} refers to the expectation of {X_{n}} under the initial
state X_{0} has distribution p(.),
and s^{2} is the asymptotic variance of S_{n}$/\sqrt{n}$.
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 P_{p}
refers to the probability of {X_{n}} under the initial
state X_{0} 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.