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Contents of PMS, Vol. 28, Fasc. 2,
pages 281 - 304
 

SMALL DEVIATION OF SUBORDINATED PROCESSES OVER COMPACT SETS

Werner Linde
Pia Zipfel

Abstract: Let A = (A(t))
         t>0  be a subordinator. Given a compact set K < [0, oo ) we prove two-sided estimates for the covering numbers of the random set (A(t) : t  (-  K) which depend on the Laplace exponent P of A and on the covering numbers of K . This extends former results in the case K = [0,1] . Using this we find the behavior of the small deviation probabilities for subordinated processes (    (   ))
 WH   A(t)  t (- K  , where WH  is a fractional Brownian motion with Hurst index 0 < H < 1 . The results are valid in the quenched as well as in the annealed case. In particular, those questions are investigated for Gamma processes. Here some surprising new phenomena appear. As application of the general results we find the behavior of logP (supt (- K |Za(t)|< e) as e --> 0 for the a -stable Lévy motion Za  . For example, if K is a self-similar set with Hausdorff dimension D > 0 , then this behavior is of order - e-aD  in complete accordance with the Gaussian case a = 2 .

2000 AMS Mathematics Subject Classification: Primary: 60G51; Secondary: 60G15, 60G52, 28A80, 60G18.

Keywords and phrases: Subordinator, fractional Brownian motion, covering numbers, Gamma process, a -stable Lévy motion.

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