Convolution tails, product tails and domains of attraction. (English) Zbl 0577.60019

Cline, Daren B. H.

Probab. Theory Relat. Fields 72, 529-557 (1986).

A distribution function is said to have an exponential tail \(\bar F(t)=F(t,\infty)\) if \(e^{\alpha u}\bar F(t+u)\) is asymptotically equivalent to \(\bar F(t),\) \(t\to \infty\), for all u. In this case \(\bar F(\ln t)\) 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 \(\bar H\) and its components \(\bar F\) and \(\bar G,\) providing conditions for \(\lim \bar H/\bar F\) to exist. In addition, we are able to describe the asymptotic nature of \(\bar H\) 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 \(H_ 1=F_ 1*G_ 1\) when \(\bar F\) is asymptotically equivalent to \(\bar F\) and \(\bar G\) is equivalent to \(\bar G_ 1\). 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.

MSC:

60E07	Infinitely divisible distributions; stable distributions
60F05	Central limit and other weak theorems

Keywords:

exponential tail; domain of attraction; convolution tails; regular variation; stable domain of attraction; extreme values

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