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Lower limits and upper limits for tails of random sums supported on \(\mathbb R\). (English) Zbl 1196.60027

Summary: Based on W. Rudin [Ann. Probab. 1, 982–994 (1973; Zbl 0303.60014)], S. Foss and D. Korshunov [Ann. Probab. 35, No. 1, 366–383 (2007; Zbl 1129.60014)] and Denisov et al. [Stat. Probab. Lett. 78, No. 17, 3023–3028 (2008; Zbl 1157.60014)], we study the lower limits and upper limits of the quotients of tails \(\frac{\overline {F^{*\tau }}(x)}{\bar F(x)}\) as \(x\rightarrow \infty \), where \(\tau \) is a non-negative integer-valued random variable and \(F\) is a distribution supported on \(\mathbb R\). Some of the new results, which are different from the corresponding results of \(\mathbb R^+\), give a positive answer to Problem 2 of T. Watanabe [Probab. Theory Relat. Fields 142, No. 3–4, 367–397 (2008; Zbl 1146.60014)] under certain conditions. In addition, we give some corresponding results for the local versions and density versions.

MSC:

60E05 Probability distributions: general theory
60F99 Limit theorems in probability theory
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