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

For a distribution F*t of a random sum St=¿1+¿+¿t of i.i.d. random variables with a common distribution F on the half-line [0, 8), we study the limits of the ratios of tails as x¿8 (here, t is a counting random variable which does not depend on {¿n}n=1). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes