## 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
Full Text:

### References:

 [1] Asmussen, S. (2000). Ruin Probabilities . River Edge, NJ: World Scientific. · Zbl 0960.60003 [2] Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Probab. 16 489-518. · Zbl 1033.60053 [3] Athreya, K. and Ney, P. (1972). Branching Processes . New York: Springer. · Zbl 0259.60002 [4] Borovkov, A.A. (1976). Stochastic Processes in Queueing Theory . New York: Springer. · Zbl 0319.60057 [5] Chistyakov, V.P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Probab. Appl. 9 640-648. · Zbl 0203.19401 [6] Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Anal. Math. 26 255-302. · Zbl 0276.60018 [7] Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Probab. 1 663-673. · Zbl 0387.60097 [8] Cline, D. (1987). Convolutions of distributions with exponential and subexponential tails. J. Aust. Math. Soc. 43 347-365. · Zbl 0633.60021 [9] Denisov, D., Foss, S. and Korshunov, D. (2007). Lower limits for distributions of randomly stopped sums. Theory Probab. Appl. · Zbl 1166.60026 [10] Embrechts, P. and Goldie, C.M. (1982). On convolution tails. Stochastic Process. Appl. 13 263-278. · Zbl 0487.60016 [11] Embrechts, P., Goldie, C.M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49 335-347. · Zbl 0397.60024 [12] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance . Berlin: Springer. · Zbl 0873.62116 [13] Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55-72. · Zbl 0518.62083 [14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications , 2 . New York: Wiley. · Zbl 0219.60003 [15] Foss, S. and Korshunov, D. (2007). Lower limits and equivalences for convolution tails. Ann. Probab. 35 366-383. · Zbl 1129.60014 [16] Harris, T. (1963). The Theory of Branching Processes . Berlin: Springer. · Zbl 0117.13002 [17] Kalashnikov, V. (1997). Geometric Sums : Bounds for Rare Events with Applications. Risk Analysis , Reliability , Queueing . Dordrecht: Kluwer. · Zbl 0881.60043 [18] Korshunov, D. (1997). On distribution tail of the maximum of a random walk. Stochastic Process. Appl. 27 97-103. · Zbl 0942.60018 [19] Pakes, A.G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 407-424. · Zbl 1051.60019 [20] Rogozin, B.A. (2000). On the constant in the definition of subexponential distributions. Theory Probab. Appl. 44 409-412. · Zbl 0971.60009 [21] Rudin, W. (1973). Limits of ratios of tails of measures. Ann. Probab. 1 982-994. · Zbl 0303.60014 [22] Shimura, T. and Watanabe, T. (2005). Infinite divisibility and generalized subexponentiality. Bernoulli 11 445-469. · Zbl 1081.60016 [23] Teugels, J.L. (1975). The class of subexponential distributions. Ann. Probab. 3 1000-1011. · Zbl 0374.60022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.