Tail Behavior of Randomly Weighted Sums

Abstract

Let {X t , t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θ t , t ≥ 1} be a sequence of positive random variables independent of the sequence {X t , t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X (∞) = ∑ t=1 ∞Θ t X t + (where X + = max{0, X}) and max1≤k<∞∑ t=1 k Θ t X t , 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 X 1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X 1, X 2,… as independent and identically distributed positive random variables. If X 1 has a regularly varying tail of index -α, where α > 0, and if {Θ t , t ≥ 1} is a positive sequence of random variables independent of {X t }, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θ t , t ≥ 1}, X (∞) = ∑ t=1 ∞Θ t X t 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 X 1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑ t=1 ∞Θ t 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.