Electronic Communications in Probability

On the infinite sums of deflated Gaussian products

Enkelejd Hashorva, Lanpeng Ji, and Zhongquan Tan

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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

Permanent link to this document
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

Keywords
Gaussian products infinite sums random deflation exact tail asymptotics max-domain of attraction regular variation chi-square distribution

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

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


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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.
  • Dobrić, 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; Klüppelberg, 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)