Limsup behaviors of multi-dimensional selfsimilar processes with independent increments

Abstract

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 $\underset{t\to \infty }{\lim }g(t)=\infty$, there is $C\in [0,\infty ]$ such that $\mathrm{limsup}|X(t)|/(t^{H}g(|\log t|))=C$ 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.

40 pages

References in corpus (1)

• Exact Hausdorff measure on the boundary of a Galton--Watson tree