The growth of additive processes

TL;DR

This paper investigates the small-time growth behavior of additive processes in , establishing conditions under which their scaled supremum converges to zero or infinity, generalizing known results for Lévy processes.

Contribution

It extends the understanding of the asymptotic growth of additive processes by identifying conditions that make the growth bounds sharp, generalizing Pruitt's results for Lévy processes.

Findings

Characterization of the  additive process growth via indices , 

Conditions for the sharpness of the growth bounds

Extension of Pruitt's results to non-Le9vy additive processes

Abstract

Let $X_t$ be any additive process in $\mathbb{R}^d.$ There are finite indices $\delta_i, \beta_i, i=1,2$ 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/\eta}X_t^*= \cases{0, \quad if $\eta>\delta_1$, \cr\infty, \quad if $\eta<\delta_2$,} \limsup_{t\to0}u(t)^{-1/\eta}X_t^*= \cases{0, \quad if $\eta>\beta_2$, \cr\infty, \quad if $\eta<\beta_1$,}\qquad {a.s.}, where $X_t^*=\sup_{0\le s\le t}|X_s|.$ When $X_t$ is a L\'{e}vy process with $X_0=0$, $\delta_1=\delta_2$, $\beta_1=\beta_2$ and $u(t)=t.$ This is a special case obtained by Pruitt. When $X_t$ is not a L\'{e}vy process, its characteristics are complicated functions of $t$. However, there are interesting conditions under which $u$ becomes sharp to achieve $\delta_1=\delta_2$, $\beta_1=\beta_2.$