Classes of distributions are defined in terms of antistarshapedness or dispersiveness. Necessary and sufficient conditions are given for these classes to be closed under mixtures. These conditions characterize distributions with log-concave densities.

This study is concerned with finite Markov decision processes whose dynamics and reward structure are unknown but the state is observable exactly.

We establish a learning algorithm which yields an optimal policy and construct an adaptive policy which is optimal under the average expected reward criterion.

Continuous random search methods with an average complexity given by O(log(1/ε)) for ε → 0 where ε is a given accuracy were presented in a recent paper. In this article an example of an O(log log(1/ε)) method is presented and illustrated.

A system is subject to a sequence of shocks occurring randomly at times n = 1, 2, ···; each shock causes a random amount of damage. The system might fail at any point in time n, and the probability of a failure depends on the history of the system. Upon failure the system is replaced by a new and identical system and a cost is incurred. If the system is replaced before failure a smaller cost is incurred. We study the problem of specifying a replacement rule which minimizes the long-run (expected) average cost per unit time. A special case, in which the system fails when the total damage first exceeds a fixed threshold, is analysed in detail.