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

(i) ${X}_{1},{X}_{2},\cdots$ are conditionally independent given a finite state irreducible aperiodic Markov chain $\left\{{s}_{i}\right\}$,

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

##### MSC:
 60F05 Central limit and other weak theorems 60G70 Extreme value theory; extremal processes (probability theory) 60K15 Markov renewal processes