ELSEVIER Statistics & Probability Letters 32 (1997) 125-131 STATISTICal & A local limit theorem for hidden Markov chains Michael Maxwell *, Michael Woodroofe Department of Mathematics, University of Michioan, Ann Arbor, MI 48109-1003, USA Received September 1995; revised February 1996 Abstract A local limit theorem is proved for partial sums of a hidden Markov chain, assuming global asymptotic normality for a related sum, a fairly weak mixing condition, and a non-lattice condition. The proof proceeds by a study of the conditional characteristic functions, the analysis of which relies heavily on a theorem from Breiman (1968). The paper concludes with a Cesaro type limit theorem for the joint distributions of the Markov chain and the partial sums. Keywords: Partial sum process; Stationary sequence; Ergodic theorem 1. Introduction The global central limit theorem for normalized sums of dependent random variables has been investigated extensively, especially under the assumption of stationarity. See, e.g., the review by Peligrad (1986). Much less is known, however, about local limit theorems. See Wang (1990) for a recent contribution and further references. In this paper sufficient conditions are obtained for a local limit theorem to hold when the sequence forms a stationary hidden Markov chain. Let Wo, W1 .... be a strictly stationary ergodic Markov chain. Let X1, X2 .... denote a sequence of real random variables which are conditionally independent given W = (W0, W1 .... ) and for which the conditional distribution of Xk given the entire sequence depends only on Wk. In this case )(1, )(2 .... is called a hidden Markov chain. The probability space on which the Wk and Xk are defined is denoted by (f2,~C,P) and the initial stationary distribution by rq i.e., P = Pn. Assume EXI = 0 and 0 < EX 2 < cx~, and let Sn = Xl + -.. + An. The main result of this paper is that under quite modest conditions and x/nP(a < Sn <~b)---*c(b- a) n - E v/-ke(Wk E A, a < Sk <~b)--* crc(A)(b-a) n k=l (1) (2) * Corresponding author. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PII S0167-7152(96)00064-8