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Tail Behavior of Randomly Weighted Sums

Part of: Nonparametric inference Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Rajat Subhra Hazra  and
Krishanu Maulik
Rajat Subhra Hazra*
Affiliation:
Indian Statistical Institute
Krishanu Maulik*
Affiliation:
Indian Statistical Institute
*
∗ Current address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland. Email address: rajatmaths@gmail.com
∗∗ Postal address: Statistics and Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India. Email address: krishanu@isical.ac.in
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Abstract

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Let {Xt, t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θt, t ≥ 1} be a sequence of positive random variables independent of the sequence {Xt, t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X(∞) = ∑t=1∞ΘtXt+ (where X+ = max{0, X}) and max1≤k<∞∑t=1kΘtXt, and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X1, X2,… as independent and identically distributed positive random variables. If X1 has a regularly varying tail of index -α, where α > 0, and if {Θt, t ≥ 1} is a positive sequence of random variables independent of {Xt}, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θt, t ≥ 1}, X(∞) = ∑t=1∞ΘtXt converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X(∞) has a regularly varying tail then does X1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑t=1∞Θt along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

Keywords

Regular variationheavy tailasymptotic independenceBreiman's theoremproduct of random variablessubexponential

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 794 - 814
Copyright
© Applied Probability Trust 

References

Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.CrossRefGoogle Scholar
Chen, Y., Ng, K. W. and Tang, Q. (2005). Weighted sums of subexponential random variables and their maxima. Adv. Appl. Prob. 37, 510–522.Google Scholar
Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.Google Scholar
Davis, R. A. and Resnick, S. I. (1996). Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Prob. 6, 1191–1210.CrossRefGoogle Scholar
Denisov, D. and Zwart, B. (2007). On a theorem of Breiman and a class of random difference equations. J. Appl. Prob. 44, 1031–1046.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.Google Scholar
Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-tailed and Subexponential Distributions. Springer, New York.Google Scholar
Hult, H. and Samorodnitsky, G. (2008). Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14, 838–864.Google Scholar
Jacobsen, M., Mikosch, T., Rosiński, J. and Samorodnitsky, G. (2009). Inverse problems for regular variation of linear filters, a cancellation property for σ-finite measures and identification of stable laws. Ann. Appl. Prob. 19, 210–242.Google Scholar
Jessen, A. H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80, 171–192.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.CrossRefGoogle Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269.Google Scholar
Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187.Google Scholar
Ledford, A. W. and Tawn, J. A. (1997). Modelling dependence within Joint tail regions. J. R. Statist. Soc. Ser. B 59, 475–499.Google Scholar
Pratt, J. W. (1960). On interchanging limits and integrals. Ann. Math. Statist. 31, 74–77.Google Scholar
Resnick, S. I. (2007). Heavy-tail Phenomena. Springer, New York.Google Scholar
Resnick, S. I. and Willekens, E. (1991). Moving averages with random coefficients and random coefficient autoregressive models. Commun. Statist. Stoch. Models 7, 511–525.Google Scholar
Wang, D. and Tang, Q. (2006). Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, 253–272.CrossRefGoogle Scholar
Zhang, Y., Shen, X. and Weng, C. (2009). Approximation of the tail probability of randomly weighted sums and applications. Stoch. Process. Appl. 119, 655–675.Google Scholar