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Laws of the iterated logarithm of "limsup" type are studied formulti-dimensional selfsimilar processes $\{X(t)\}$ with independent incrementshaving 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 theexistence of $g(t)$ with C=1 is obtained. In the case where $g(t)$ with C=1does not exist, a criterion to classify functions $g(t)$ according to C=0 or$C=\infty$ is given. Moreover, various "limsup" type laws with identificationof the positive constants $C$ are explicitly presented in several propositionsand examples. The problems that exchange the roles of $\{X(t)\}$ and $g(t)$ arealso discussed.

Limsup behaviors of multi-dimensional selfsimilar processes with independent increments