m at h . PR ] 3 1 Ja n 20 07 On lower limits and equivalences for distribution tails of randomly stopped sums

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 an independent counting random variable). We also consider applications of obtained results to random walks, compound Poisson distributions, infinitely divisible laws, and sub-critical branching processes. Let τ be a counting random variable which does not depend on {ξ n } n≥1 and has finite mean. Denote by F * τ the distribution of a randomly stopped sum S τ = ξ 1 +. .. + ξ τ. In this paper we discuss how does the tail behaviour of F * τ relate to that of F and, in particular, under what conditions