Limsup behaviors of multi-dimensional selfsimilar processes with independent increments

TL;DR

This paper investigates the asymptotic behavior of multi-dimensional self-similar processes with independent increments, establishing laws of the iterated logarithm and conditions for specific limsup constants.

Contribution

It provides new necessary and sufficient conditions for limsup behaviors of self-similar processes, including explicit laws and classifications based on function growth.

Findings

Established limsup laws with explicit constants for self-similar processes.

Derived conditions for the existence of functions with specific limsup constants.

Classified functions according to their limsup behavior in the processes.

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 $\displaystyle \lim_{t\to\infty}g(t) = \infty$, 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.