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