Advances in Applied Probability

Tail behavior of randomly weighted sums

Rajat Subhra Hazra and Krishanu Maulik

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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.

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 3 (2012), 794-814.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1346955265

Digital Object Identifier
doi:10.1239/aap/1346955265

Mathematical Reviews number (MathSciNet)
MR3024610

Zentralblatt MATH identifier
1264.60036

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference

Keywords
Regular variation heavy tail asymptotic independence Breiman's theorem product of random variables subexponential

Citation

Hazra, Rajat Subhra; Maulik, Krishanu. Tail behavior of randomly weighted sums. Adv. in Appl. Probab. 44 (2012), no. 3, 794--814. doi:10.1239/aap/1346955265. https://projecteuclid.org/euclid.aap/1346955265


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