  We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain
$(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show
for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only
possible weak limit and provide necessary and sufficient conditions for the
convergence. More generally, we also consider the weak convergence of $tL(Y_t)$
as $t\to0$ for a decreasing function $L$ that is slowly varying at zero.
Various examples demonstrating the applicability of the results are presented.
