Sinai's condition for real valued L\'evy processes
Abstract
We prove that the upward ladder height subordinator H associated to a real valued L\'evy process has Laplace exponent φ that varies regularly at ∞ (resp. at 0) if and only if the underlying L\'evy process satisfies Sinai's condition at 0 (resp. at ∞). Sinai's condition for real valued L\'evy processes is the continuous time analogue of Sinai's condition for random walks. We provide several criteria in terms of the characteristics of to determine whether or not it satisfies Sinai's condition. Some of these criteria are deduced from tail estimates of the L\'evy 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 Gr\"ubel
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.