Convolution tails, product tails and domains of attraction

Abstract

A distribution function is said to have an exponential tail F(t) = F(t, ∞) if e αu F(t+u) is asymptotically equivalent to F(t), t→∞, t→∞, for all u. In this case F(lnt) is regularly varying. For two such distributions, F and G, the convolution H=F*G also has an exponential tail. We investigate the relationship between H and its components F and G, providing conditions for lim H/F to exist. In addition, we are able to describe the asymptotic nature of H when the limit is infinite, for many cases. This corresponds to determining both the domain of attraction and the norming constants for the product of independent variables whose distributions have regularly varying tails. In addition, we compare the tails of H=F*G with H 1=F 1*G 1when F is asymptotically equivalent to F and G is equivalent to G 1. Such a comparison corresponds to the “balancing” consideration for the product of independent variables in stable domains of attraction. We discover that there are several distinct comparisons possible.