The growth of additive processes

Abstract

Let $X_t$ be any additive process in $\mathbb{R}^d.$ There are finite indices $δ_i, β_i, i=1,2$ and a function $u$, all of which are defined in terms of the characteristics of $X_t$, such that \liminf_{t\to0}u(t)^{-1/η}X_t^*= \cases{0, \quad if $η>δ_1$, \cr\infty, \quad if $η<δ_2$,} \limsup_{t\to0}u(t)^{-1/η}X_t^*= \cases{0, \quad if $η>β_2$, \cr\infty, \quad if $η<β_1$,}\qquad {a.s.}, where $X_t^*=\sup_{0\le s\le t}|X_s|.$ When $X_t$ is a Lévy process with $X_0=0$, $δ_1=δ_2$, $β_1=β_2$ and $u(t)=t.$ This is a special case obtained by Pruitt. When $X_t$ is not a Lévy process, its characteristics are complicated functions of $t$. However, there are interesting conditions under which $u$ becomes sharp to achieve $δ_1=δ_2$, $β_1=β_2.$