The growth of additive processes

Abstract

Let X_t be any additive process in 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 $$\begin{aligned}\liminf_{t\rightarrow0}u(t)^{-1/\eta}X_{t}^{*}=\cases{0,\phantom{\infty,\quad\!\!\!\!\!\!}\mbox{if }\eta>\delta_{1},\cr\infty,\quad \mbox{if }\eta < \delta_{2},}\\ \limsup_{t\rightarrow0}u(t)^{-1/\eta}X_{t}^{*}=\cases{0,\phantom{\infty,\quad\!\!\!\!\!\!}\mbox{if }\eta>\beta_{2},\cr\infty,\quad \mbox{if }\eta < \beta_{1},}\end{aligned}\qquad \mbox{a.s.},$$ where X_t^*=sup _0≤s≤t|X_s|. When X_t is a Lévy process with X₀=0, δ₁=δ₂, β₁=β₂ 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 δ₁=δ₂, β₁=β₂.