We study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.

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We study the comparability of the lifetimes of heterogeneous parallel systems with independent exponentially distributed components. It is known that the order statistics of systems composed of two types of components may be comparable with respect to the star transform order. On what concerns the stronger convex transform order, results have been obtained only for the sample maxima assuming that one of the systems is homogeneous. We prove, under the same assumptions as for the star transform ordering, that the lifetimes of heterogeneous parallel systems are not comparable with respect to the convex transform order.

Conditional N-demimartingales Maximal inequalities Moment inequalities Strong law of large numbers Conditional complete convergence a b s t r a c tIn this paper we deal with the classes of F -demimartingales and conditional N-demimartingales and for these new classes of random objects we provide a number of maximal and moment inequalities as well as related asymptotic results.

The class of N-demimartingales generalizes in a natural way the concept of negative association and includes as special cases martingales with respect to the natural choice of σ-algebras. For this class of random variables, a number of maximal and other inequalities were obtained by Christofides (2003) and Prakasa Rao (2004, 2007. In this paper we prove Azuma's inequality for N-demimartingales and as a corollary we obtain an exponential inequality for negatively associated random variables.