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On the supremum of an infinitely divisible process. (English) Zbl 0633.60025
A distribution function F is said to be long-tailed if we have \[ \lim_{x\to \infty}(1-F(x-y))/(1-F(x))=1 \] for all \(y\in {\mathbb{R}}\). Let \(X=\{X_ t\}_{t\geq 0}\) be an infinitely divisible process and define \(Y_ t=\sup_{0\leq s\leq t}X_ s\) for all \(t\geq 0.\)
The principal result of this paper, which generalizes a result of S. M. Berman [ibid. 23, 281-290 (1986; Zbl 0612.60063)], asserts that the distribution of each \(X_ t\) is long-tailed if and only if the distribution of each \(Y_ t\) is long-tailed, in which case, for each t, the probabilities \(P(X_ t\geq u)\) and \(P(Y_ t\geq u)\) are asymptotic as \(u\to \infty\).
Reviewer: B.K.Horkelheimer

MSC:
60E07 Infinitely divisible distributions; stable distributions
60F10 Large deviations
60J99 Markov processes
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[1] Berman, S.M., The supremum of a process with stationary independent and symmetric increments, (), 281-290 · Zbl 0612.60063
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