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March 2007 The growth of additive processes
Ming Yang
Ann. Probab. 35(2): 773-805 (March 2007). DOI: 10.1214/009117906000000593

Abstract

Let Xt be any additive process in ℝd. There are finite indices δi, βi, i=1, 2 and a function u, all of which are defined in terms of the characteristics of Xt, 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 Xt*=sup 0≤st|Xs|. When Xt is a Lévy process with X0=0, δ1=δ2, β1=β2 and u(t)=t. This is a special case obtained by Pruitt. When Xt 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.

Citation

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Ming Yang. "The growth of additive processes." Ann. Probab. 35 (2) 773 - 805, March 2007. https://doi.org/10.1214/009117906000000593

Information

Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1117.60049
MathSciNet: MR2308597
Digital Object Identifier: 10.1214/009117906000000593

Subjects:
Primary: 60F15 , 60G51
Secondary: 60E07 , 60G17

Keywords: Additive processes , growth indices , short-term behavior of X_t^*

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 2 • March 2007
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