product tails and domains of attraction Convolution tails

A distribution function is said to have an exponential tail F(t) = F(t, ) if euF(t+u) is asymptotically equivalent to F(t), t, t, for all u. In this case F(ln t) 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 H1=F1*G1when F is asymptotically equivalent to F and G is equivalent to G1. 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. 1986 Springer-Verlag.

Convolution tails, product tails and domains of attraction Academic Article