Large deviations and laws of the iterated logarithm for the local times of additive stable processes

Abstract

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

η^x([0, t]^p)=∫₀^t⋯∫₀^tδ_x(X₁(s₁)+⋯+X_p(s_p)) ds₁⋯ ds_p

of the multiparameter process X̅(t₁, …, t_p), and for its supremum norm sup _x∈ℝ^dη^x([0, t]^p). Our results apply to the law of the iterated logarithm and our approach is based on Fourier analysis, moment computation and time exponentiation.