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 $X_1(t),...,X_p(t)$ be independent, d-dimensional symmetric stable processes with stable index $0<\alpha \le 2$ and consider the additive stable process $\bar{X}(t_1,...,t_p)=X_1(t_1)+...+X_p(t_p)$. Under the condition $d<\alpha p$, we obtain a precise form of the large deviation principle for the local time \[η^x([0,t]^p)=\int_0^t...\int_0^tδ_x\bigl(X_1(s_1)+... +X_p(s_p)\bigr) ds_1... ds_p\] of the multiparameter process $\bar{X}(t_1,...,t_p)$, and for its supremum norm ${\sup }_{x\in {\mathbb{R}}^d}{\eta }^x([0,t{]}^p)$. Our results apply to the law of the iterated logarithm and our approach is based on Fourier analysis, moment computation and time exponentiation.

Published at http://dx.doi.org/10.1214/009117906000000601 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm
• Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks
• Large deviations for renormalized self-intersection local times of stable processes

Cited by in corpus (2)

• Large Deviations for Brownian Intersection Measures
• Moderate deviations and laws of the iterated logarithm for the local times of additive Lévy processes and additive random walks