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Limit distributions of maximal segmental score among Markov-dependent partial sums. (English) Zbl 0767.60017

Let S n denote the n-th partial sum of a sequence {X i } of random variables, and let M(n)=sup 0kln (S l -S k ). When X 1 ,X 2 , are independent identically distributed bounded nonlattice random variables satisfying EX i <0 and P(X i >0)>0, D. Iglehart [Ann. Math. Stat. 43, 627-635 (1972; Zbl 0238.60072)] proved that M(n)-(lnn)/θ has a limiting double exponential distribution, where θ>0 satisfies E(e θX )=1. In the present paper the authors obtain a similar result for a sequence {X i } satisfying the following conditions:

(i) X 1 ,X 2 , are conditionally independent given a finite state irreducible aperiodic Markov chain {s i },

(ii) the conditional distribution of X i given {s i } depends only on the two states s i and s i+1 .


MSC:
60F05Central limit and other weak theorems
60G70Extreme value theory; extremal processes (probability theory)
60K15Markov renewal processes