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Saddlepoint expansions for sums of Markov dependent variables on a continuous state space. (English) Zbl 0723.60019

Based on the conjugate kernel studied in I. Iscoe, P. Ney and E. Nummelin [Adv. Appl. Math. 6, 373-412 (1985; Zbl 0602.60034)] we derive saddlepoint expansions for either the density or distribution function of a sum \(f(X_ 1)+...+f(X_ n)\), where the \(X_ i's\) constitute a Markov chain. The chain is assumed to satisfy a strong recurrence condition which makes the results here very similar to the classical results for i.i.d. variables. In particular we establish also conditions under which the expansions hold uniformly over the range of the saddlepoint. Expansions are also derived for sums of the form \(f(X_ 1,X_ 0)+f(X_ 2,X_ 1)+...+f(X_ n,X_{n-1})\) although the uniformity result just mentioned does not generalize.

MSC:

60E99 Distribution theory
60F99 Limit theorems in probability theory

Citations:

Zbl 0602.60034
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References:

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