Tail Behavior of Randomly Weighted Sums

TL;DR

This paper investigates the tail behavior and convergence of randomly weighted sums of heavy-tailed variables, providing new sufficient conditions involving slowly varying functions and Mellin transforms.

Contribution

It introduces novel conditions for tail probability control and convergence of weighted sums, extending prior results to broader settings with less restrictive moment assumptions.

Findings

Conditions on slowly varying functions influence tail behavior.

Convergence of sums depends on moment conditions of weights.

Mellin transform conditions are necessary for tail equivalence.

Abstract

Let $\{X_t, t \geq 1\}$ be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails and $\{ \Theta_t, t\geq1 \}$ be a sequence of positive random variables independent of the sequence $\{X_t, t \geq 1\}$. We shall discuss the tail probabilities and almost sure convergence of $\xinf=\sum_{t=1}^{\infty}\Theta_t X_t^{+}$ (where $X^+=\max\{0,X\}$) and $\max_{1\leq k<\infty} \sum_{t=1}^{k}\Theta_t X_t$ 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 $X_1$ helps 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 regularly varying tail of index $-\alpha$, where $\alpha>0$, and if $\{\Theta_t,t\geq1\}$ is a positive sequence of random variables independent of $\{X_t\}$, then it is known, which can also be obtained from the sufficient conditions above, that under some appropriate moment conditions on $\{\Theta_t,t\geq1\}$, $\xinf=\sum_{t=1}^{\infty}\Theta_t X_t$ converges with probability 1 and has regularly varying tail of index $-\alpha$. Motivated by the converse problems in Jacobsen et al. (2009) we ask the question that, if $\xinf$ has regularly varying tail, then does $X_1$ have regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including nonvanishing Mellin transform of $\sum_{t=1}^\infty \Theta_t$ 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.