On the small-time behavior of subordinators
Abstract
We prove several results on the behavior near t=0 of
Y
−t
t
for certain
(0,∞)
-valued stochastic processes
(
Y
t
)
t>0
. In particular, we show for Lévy subordinators that the Pareto law on
[1,∞)
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→0
for a decreasing function
L
that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.