Limsup behaviors of multi-dimensional selfsimilar processes with independent increments

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 ${\displaystyle \underset{t\to \mathrm{\infty }}{\lim }g(t)=\mathrm{\infty }}$, there is $C\in [0,\infty]$suchthat$\limsup|X(t)|/(t^Hg(|\log t|))= C$a\mathrm{.}s\mathrm{.}as$t \to\infty $,inaddition,as$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 $C=\mathrm{\infty }$ 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.