Mathematics – Probability
Scientific paper
2006-03-07
Annals of Probability 2007, Vol. 35, No. 2, 602-648
Mathematics
Probability
Published at http://dx.doi.org/10.1214/009117906000000601 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins
Scientific paper
10.1214/009117906000000601
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 \[\eta^x([0,t]^p)=\int_0^t...\int_0^t\delta_x\bigl(X_1(s_1)+... +X_p(s_p)\bigr) ds_1... ds_p\] 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.
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