# zbMATH — the first resource for mathematics

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
##### Keywords:
long-tailed; infinitely divisible process
Full Text:
##### References:
 [1] Berman, S.M., The supremum of a process with stationary independent and symmetric increments, (), 281-290 · Zbl 0612.60063 [2] Berman, S.M., Limit theorems for sojourns of stochastic processes, (), 40-71 [3] Embrechts, P.; Goldie, C.M.; Veraverbeke, N., Subexponentiality and infinite divisibility, Z. wahrsch. verw. geb., 49, 335-347, (1979) · Zbl 0397.60024 [4] Feller, W., () [5] Sgibnev, M.S., Infinitely divisible distributions belonging to the class $$L$$(γ), Sibirsk mat. zh., (1987), to appear in [6] Wilekens, E., Higher order theory for subexponential distributions (in Dutch), (), K.U. Leuven [7] Willekens, E., Subexponentiality on the real line, Technical report, (1986), K.U. Leuven
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.