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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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