A bivariate exponential distribution due to Gumbel and a bivariate Lomax distribution due to Lindley and Singpurwalla have been characterized through a bivariate extension of a univariate characterizing property of the exponential and the Lomax distribution; the characterization is based on mean residual lives and hazard rates.

A distribution function F on (0,∞) belongs to the subexponential class if the ratio of 1 – F (2)(x) to 1 – F(x) converges to 2 as x →∞. A necessary condition for membership in is used to prove that a certain class of functions previously thought to be contained in has members outside of . Sufficient conditions on the tail of F are found which ensure F belongs to ; these conditions generalize previously published conditions.

Invariant integrals of a Lotka-Volterra system with infinitely many species are introduced. The values of these integrals are given by the probabilities of certain configurations of random points on a circle when the probability density on the circle satisfies a certain symmetry condition.

We prove existence and uniqueness of a stationary regime for a model of a circuit-switching network with infinitely many nodes.