Sinai's condition for real valued Lévy processes

Victor Rivero (1)

1 MODAL'X - Modélisation aléatoire de Paris X

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 $\infty$ (resp. at $0$) if and only if the underlying Lévy process $\xi$ satisfies Sinai's condition at $0$ (resp. at $\infty$). Sinai's condition for real valued Lévy processes is the continuous time analogue of Sinai's condition for random walks. We provide several criteria in terms of the characteristics of $\xi$ to determine whether or not it satisfies Sinai's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of $H,$ here obtained, and which are analogous to the estimates of the tail distribution of the ladder height random variable of a random walk which are due to Veraverbeke and Grübel

Mots clés: Lévy processes; Fluctuation theory; Regular Variation; long tailed Lévy measures

Domaines: Probabilités [math.PR]

Victor Rivero. Sinai's condition for real valued Lévy processes. 2005. ⟨hal-00004962⟩