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
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\[\liminf_{t\rightarrow0}u(t)^{-1/\eta}X_{t}^{*}=\cases{0,\phantom{\infty,\quad\!\!\!\!\!\!}\mbox{if }\eta>\delta_{1},\cr\infty,\quad \mbox{if }\eta\textless \delta_{2},}\]
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\[\limsup_{t\rightarrow0}u(t)^{-1/\eta}X_{t}^{*}=\cases{0,\phantom{\infty,\quad\!\!\!\!\!\!}\mbox{if }\eta>\beta_{2},\cr\infty,\quad \mbox{if }\eta\textless \beta_{1},}\qquad \mbox{a.s.},\]
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where Xt*=sup 0≤s≤t|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.