We study the upper tail behaviors of the local times of the additive stable processes. Let X1(t), · · · , Xp(t) be independent, d-dimensional symmetric stable processes with stable index 0 < α ≤ 2 and consider the additive stable process X(t1, · · · , tp) = X1(t1) + · · · + Xp(tp). Under the condition d < αp, we obtain a precise form of large deviation principle for the local time

Let (X n) be a recurrent Markov chain on Z 2 with X 0 = (0; 0) such that for some constant C, PX k = (0; 0)] C k , and whose truncated Green function is slowly varying at innnity. Let L 0 n denote the local time at zero of such a Markov chain. We prove various moderate and large deviation statements and limit laws for rescaled versions of L 0 n , including functional versions of these. A versio...

We approximate the Bolker-Pacala model of population dynamics with the logistic Markov chain and analyze the latter. We find the asymptotics of the degenerated hypergeometric function and use these to prove a local CLT and large deviations result. We also state global limit theorems and obtain asymptotics for the first passage time to the boundary of a large interval.