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 as x→∞ (here, τ is a counting random variable which does not depend on {ξn}n≥1). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

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