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On lower limits and equivalences for distribution tails of randomly stopped sums. (English) Zbl 1157.60315

Summary: For a distribution \(F^{*\tau}\) of a random sum \(S_\tau =\xi _{1}+ \cdots +\xi _{\tau}\) of i.i.d. random variables with a common distribution \(F\) on the half-line [\(0, \infty \)), we study the limits of the ratios of tails \(\overline{F^{*\tau}}(x)/\overline F(x)\) as \(x\rightarrow \infty \) (here, \(\tau \) is a counting random variable which does not depend on \(\{\xi _n\}_{n\geq 1}\)). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
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