## On convolution equivalence with applications.(English)Zbl 1114.60015

Summary: A distribution $$F$$ on $$(-\infty,\infty)$$ is said to belong to the class $${\mathcal S}(\gamma)$$ for some $$\gamma\geq 0$$ if $$\lim_{x\to\infty} \overline{F}(x-u)/ \overline{F}(x)= e^{\gamma u}$$ holds for all $$u$$ and $$\lim_{x\to\infty} \overline{F^{*2}}(x)/ \overline{F}(x)= 2m_F$$ exists and is finite. Let $$X$$ and $$Y$$ be two independent random variables, where $$X$$ has a distribution in the class $${\mathcal S}(\gamma)$$ and $$Y$$ is nonnegative with an endpoint $$\widehat{y}= \sup \{y: P(Y\leq y)< 1\}\in (0,\infty)$$. We prove that the product $$XY$$ has a distribution in the class $${\mathcal S}(\gamma/\widehat{y})$$. We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.

### MSC:

 60E05 Probability distributions: general theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
Full Text:

### References:

 [1] Brémaud, P. (2000) An insensitivity property of Lundbergś estimate for delayed claims. J. Appl. Probab., 37(3), 914-917. · Zbl 0968.62074 [2] 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 [3] Chover, J., Ney, P. and Wainger, S. (1973a) Functions of probability measures. J. Anal. Math., 26, 255-302. · Zbl 0276.60018 [4] Chover, J., Ney, P. and Wainger, S. (1973b) Degeneracy properties of subcritical branching processes. Ann. Probab., 1, 663-673. · Zbl 0387.60097 [5] Cline, D.B.H. (1987) Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A, 43(3), 347-365. · Zbl 0633.60021 [6] Cline, D.B.H. and Samorodnitsky, G. (1994) Subexponentiality of the product of independent random variables. Stochastic Process. Appl., 49(1), 75-98. · Zbl 0799.60015 [7] Embrechts, P. (1983) A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Probab., 20(3), 537-544. JSTOR: · Zbl 0536.60022 [8] Embrechts, P. and Goldie, C.M. (1994) Perpetuities and random equations. In P. Mandl and M. Hus?ková (eds) Asymptotic Statistics: Proceedings of the Fifth Prague Symposium, pp. 75-86. Heidelberg: Physica. [9] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag. · Zbl 0873.62116 [10] Goldie, C.M. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., 1(1), 126-166. · Zbl 0724.60076 [11] Klüppelberg, C. and Mikosch, T. (1995) Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli, 1(1-2), 125-147. · Zbl 0842.60030 [12] Klüppelberg, C., Mikosch, T. and Schärf, A. (2003) Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli, 9(3), 467-496. · Zbl 1044.60013 [13] Lund, R., McCormick, W.P. and Xiao, Y. (2004) Limiting properties of Poisson shot noise processes. J. Appl. Probab., 41(3), 911-918. · Zbl 1068.60031 [14] McCormick, W.P. (1997) Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Probab., 34(3), 643-656. JSTOR: · Zbl 0886.60041 [15] Pakes, A.G. (2004) Convolution equivalence and infinite divisibility. J. Appl. Probab., 41(2), 407-424. · Zbl 1051.60019 [16] Rogozin, B.A. (2000) On the constant in the definition of subexponential distributions. Theory Probab. Appl., 44(2), 409-412. · Zbl 0971.60009 [17] Rogozin, B.A. and Sgibnev, M.S. (1999) Banach algebras of measures on the line with given asymptotics of distributions at infinity. Siberian Math. J., 40(3), 565-576. · Zbl 0936.46021 [18] Ross, S.M. (1983) Stochastic Processes. New York: Wiley. · Zbl 0555.60002 [19] Samorodnitsky, G. (1998) Tail behaviour of some shot noise processes. In R.J. Adler, R.E. Feldman and M.S. Taqqu (eds), A Practical Guide to Heavy Tails: Statistical Techniques and Applications, pp. 473-486. Boston: Birkhäuser. · Zbl 0937.60037 [20] Shimura, T. and Watanabe, T. (2005) Infinite divisibility and generalized subexponentiality. Bernoulli, 11(3), 445-469. · Zbl 1081.60016 [21] Tang, Q. and Tsitsiashvili, G. (2003) Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process. Appl., 108(2), 299-325. · Zbl 1075.91563 [22] Tang, Q. and Tsitsiashvili, G. (2004) Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. in Appl. Probab., 36(4), 1278-1299. · Zbl 1095.91040
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.