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Limit distributions of maximal segmental score among Markov-dependent partial sums. (English) Zbl 0767.60017
Let $S\sb n$ denote the $n$-th partial sum of a sequence $\{X\sb i\}$ of random variables, and let $M(n)=\sup\sb{0\le k\le l\le n}(S\sb l-S\sb k)$. When $X\sb 1,X\sb 2,\dots$ are independent identically distributed bounded nonlattice random variables satisfying $EX\sb i<0$ and $P(X\sb i>0)>0$, {\it D. Iglehart} [Ann. Math. Stat. 43, 627-635 (1972; Zbl 0238.60072)] proved that $M(n)-(\ln n)/\theta$ has a limiting double exponential distribution, where $\theta>0$ satisfies $E(e\sp{\theta X})=1$. In the present paper the authors obtain a similar result for a sequence $\{X\sb i\}$ satisfying the following conditions: (i) $X\sb 1,X\sb 2,\dots$ are conditionally independent given a finite state irreducible aperiodic Markov chain $\{s\sb i\}$, (ii) the conditional distribution of $X\sb i$ given $\{s\sb i\}$ depends only on the two states $s\sb i$ and $s\sb{i+1}$.

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