A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

This paper is motivated by the modelling of leaching of bacteria through soil. A semi-linear process X t − may be used to describe the soil-drying process between rain showers. This is a backward recurrence time process that corresponds to the renewal process of instances of rain. If a bacterium moves according to another process h, then the fact that h(t) stays above X t − means that the bacterium never hits a dry patch of soil and so survives. We describe a critical behaviour of h that separates the cases when survival is possible with a positive probability from the cases when this probability vanishes. An explicit formula for the survival probability is obtained in case h is linear and rain showers follow a Poisson process.

Assume a given sequence of events to be strongly mixing at a polynomial or exponential rate. We show that the conclusion of the second Borel-Cantelli lemma holds if the series of the probabilities of the events diverges at a certain rate depending on the mixing rate of the events. An application to necessary moment conditions for the strong law of large numbers is given.

The general part of the Borel-Cantelli lemma says that for any sequence of events (An) defined on a probability space (Ω, Σ, P), the divergence of ΣnP(An) is necessary for P(An i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the An are independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced by for some . We will see that this property of (An) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory of ϕ -branching processes will show the tractability of the method.