Sinai's condition for real valued L\'{e}vy processes

We prove that the upward ladder height subordinator $H$ associated to a real valued L\'{e}vy process $\xi$ has Laplace exponent $\phi$ that varies regularly at $\infty$ (resp. at 0) if and only if the underlying L\'{e}vy process $\xi$ satisfies Sinai's condition at 0 (resp. at $\infty$). Sinai's condition for real valued L\'{e}vy 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 L\'{e}vy 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\"{u}bel