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September 2012 Tail behavior of randomly weighted sums
Rajat Subhra Hazra, Krishanu Maulik
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Adv. in Appl. Probab. 44(3): 794-814 (September 2012). DOI: 10.1239/aap/1346955265

Abstract

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=1ΘtXt+ (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=1ΘtXt 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=1Θt 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.

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Rajat Subhra Hazra. Krishanu Maulik. "Tail behavior of randomly weighted sums." Adv. in Appl. Probab. 44 (3) 794 - 814, September 2012. https://doi.org/10.1239/aap/1346955265

Information

Published: September 2012
First available in Project Euclid: 6 September 2012

zbMATH: 1264.60036
MathSciNet: MR3024610
Digital Object Identifier: 10.1239/aap/1346955265

Subjects:
Primary: 60G70
Secondary: 62G32

Keywords: Asymptotic independence , Breiman's theorem , heavy tail , product of random variables , regular variation , subexponential

Rights: Copyright © 2012 Applied Probability Trust

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Vol.44 • No. 3 • September 2012
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