## Electronic Communications in Probability

### On the infinite sums of deflated Gaussian products

#### Abstract

In this paper we derive the exact tail asymptotic behaviour of $S_\infty=\sum_{i=1}^\infty \lambda_i X_iY_i$, where $\lambda_i, i\ge 1,$ are non-negative square summable deflators (weights) and $X_i,Y_i, i\ge1,$ are independent standard Gaussian random variables. Further, we consider the tail asymptotics of $S_{\infty;p}=\sum_{i=1}^\infty\lambda_i X_i|Y_i|^p, p> 1$, and also discuss the influence on the asymptotic results when $\lambda_i$'s are independent random variables.

#### Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 31, 8 pp.

Dates
Accepted: 23 July 2012
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465263164

Digital Object Identifier
doi:10.1214/ECP.v17-1921

Mathematical Reviews number (MathSciNet)
MR2955496

Zentralblatt MATH identifier
1266.60069

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G15: Gaussian processes

Rights

#### Citation

Hashorva, Enkelejd; Ji, Lanpeng; Tan, Zhongquan. On the infinite sums of deflated Gaussian products. Electron. Commun. Probab. 17 (2012), paper no. 31, 8 pp. doi:10.1214/ECP.v17-1921. https://projecteuclid.org/euclid.ecp/1465263164

#### References

• Arendarczyk, Marek; DÈ©bicki, Krzysztof. Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli 17 (2011), no. 1, 194–210.
• Bovier, A.: Extreme values of random processes. Lecture Notes Technische Universität Berlin, 2005.
• DobriÄ‡, Vladimir; Marcus, Michael B.; Weber, Michel. The distribution of large values of the supremum of a Gaussian process. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). Astérisque No. 157-158 (1988), 95–127.
• Embrechts, Paul; KlÃ¼ppelberg, Claudia; Mikosch, Thomas. Modelling extremal events. For insurance and finance. Applications of Mathematics (New York), 33. Springer-Verlag, Berlin, 1997. xvi+645 pp. ISBN: 3-540-60931-8
• Hashorva, E., and Ji., L.: Asymptotics of the finite-time ruin probability for the Sparre Andersen risk model perturbed by inflated chi-process. (2012) Preprint.
• Hashorva, Enkelejd; Pakes, Anthony G.; Tang, Qihe. Asymptotics of random contractions. Insurance Math. Econom. 47 (2010), no. 3, 405–414.
• Hoeffding, W.: On a theorem of V.N. Zolotarev. Theory of Probab. Appl. IX 1, (1964), 89–91. (English translation)
• Ivanoff, B. Gail; Weber, N. C. Tail probabilities for weighted sums of products of normal random variables. Bull. Austral. Math. Soc. 58 (1998), no. 2, 239–244.
• Kallenberg, Olav. Some new representations in bivariate exchangeability. Probab. Theory Related Fields 77 (1988), no. 3, 415–455.
• Lifshits, M. A. Tail probabilities of Gaussian suprema and Laplace transform. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 2, 163–179.
• Lifshits, M. A. Gaussian random functions. Mathematics and its Applications, 322. Kluwer Academic Publishers, Dordrecht, 1995. xii+333 pp. ISBN: 0-7923-3385-3
• Lifshits, M.A. Lectures on Gaussian Processes. Springer Briefs in Mathematics, Springer, 2012.
• Liu, Yan; Tang, Qihe. The subexponential product convolution of two Weibull-type distributions. J. Aust. Math. Soc. 89 (2010), no. 2, 277–288.
• Pakes, Anthony G. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 (2004), no. 2, 407–424.
• Resnick, Sidney I. Extreme values, regular variation and point processes. Reprint of the 1987 original. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2008. xiv+320 pp. ISBN: 978-0-387-75952-4
• Zolotarev, V.M.: Concerning a certain probability problem. Theory of Probab. Appl. IX1, (1961), 201–204. (English translation)