# Convolution tail equivalent distributions: basic properties (Joint with Networks (OR) seminar).

The talk discusses the tail behaviour of 2-fold convolutions $\overline{F \star F}$ of right-unbounded distributions and to the tail behaviour of the distribution of the random sum $S_\tau$. We start with the description of all possible values of the limit of the ratio $\overline{F \star F}(x)/\overline F(x)$ as $x\to\infty$. This problem is closely related to the problem of the lower limit of this ratio and goes back to W. Rudin. The second part of the talk is devoted to conditions under which $P(S_\tau>x)\sim E\tau \overline F(x)$ as $x\to\infty$. We also consider applications of results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and branching processes.

This talk is part of the Probability series.