Limsup behaviors of multi-dimensional selfsimilar processes with independent increments

Details Limsup behaviors of multi-dimensional selfsimilar processes with independent increments Limsup behaviors of multi-dimensional selfsimilar processes with independent increments

2010-03-02

arxiv.org/abs/1003.0552v2

ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010) 79-116

Mathematics

Probability

40 pages

Scientific paper

Laws of the iterated logarithm of "limsup" type are studied for multi-dimensional selfsimilar processes $\{X(t)\}$ with independent increments having exponent $H$. It is proved that, for any positive increasing function $g(t)$ with $\displaystyle \lim_{t\to\infty}g(t) = \infty$, there is $C\in [0,\infty]$ such that $\limsup|X(t)|/(t^Hg(|\log t|))= C$ a.s. as $t \to\infty $, in addition, as $t \to 0$. A necessary and sufficient condition for the existence of $g(t)$ with C=1 is obtained. In the case where $g(t)$ with C=1 does not exist, a criterion to classify functions $g(t)$ according to C=0 or $C=\infty$ is given. Moreover, various "limsup" type laws with identification of the positive constants $C$ are explicitly presented in several propositions and examples. The problems that exchange the roles of $\{X(t)\}$ and $g(t)$ are also discussed.

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