On lower limits and equivalences for distribution tails of randomly stopped sums
Abstract
For a distribution
F
∗τ
of a random sum
S
τ
=
ξ
1
+...+
ξ
τ
of i.i.d. random variables with a common distribution
F
on the half-line
[0,∞)
, we study the limits of the ratios of tails
F
∗τ
¯
(x)/
F
¯
(x)
as
x→∞
(here,
τ
is a counting random variable which does not depend on
{
ξ
n
}
n≥1
). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.