The growth of additive processes

Details The growth of additive processes The growth of additive processes

2007-07-26

arxiv.org/abs/0707.3886v1

Annals of Probability 2007, Vol. 35, No. 2, 773-805

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000593 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000593

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 $\delta_1=\delta_2$, $\beta_1=\beta_2.$

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