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Sinaǐ’s condition for real valued Lévy processes. (English) Zbl 1115.60049
Summary: We prove that the upward ladder height subordinator \(H\) associated to a real valued Lévy process \(\xi\) has Laplace exponent \(\varphi\) that varies regularly at \(\infty\) (respectively, at 0) if and only if the underlying Lévy process \(\xi\) satisfies Sinaǐ’s condition at 0 (respectively, at \(\infty\)). Sinaǐ’s condition for real valued Lévy processes is the continuous time analogue of Sinaǐ’s condition for random walks. We provide several criteria in terms of the characteristics of \(\xi\) to determine whether or not it satisfies Sinaǐ’s condition. Some of these criteria are deduced from tail estimates of the Lévy measure of \(H\), here obtained, which are analogous to the estimates of the tail distribution of the ladder height random variable of a random walk which are due to N. Veraverbeke [Stochastic Processes Appl. 5, 27–37 (1977; Zbl 0353.60073)] and R. Grübel [J. Appl. Probab. 22, 705–709 (1985; Zbl 0574.60075)].

MSC:
60G51 Processes with independent increments; Lévy processes
60G30 Continuity and singularity of induced measures
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