On lower limits and equivalences for distribution tails of randomly stopped sums

For a distribution F*τ of a random sum Sτ = ξ1 + ... + ξτ 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. © 2008 ISI/BS.