This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

The frequency and determinants of abnormal test performance by normal individuals are critically important to clinical inference. Here we compare two approaches to predicting rates of abnormal test performance among healthy individuals with the rates actually shown by 327 neurologically normal adults aged 18–92 years. We counted how many participants produced abnormal scores, defined by three different cutoffs with test batteries of varied length, and the number of abnormal scores they produced. Observed rates generally were closer to predictions based on a series of Monte Carlo simulations than on the binomial model. They increased with the number of tests administered, decreased as more stringent cutoffs were used to identify abnormality, varied with the degree of correlation among test scores, and depended on individual differences in age, education, race, sex, and estimated premorbid IQ. Adjusting scores for demographic variables and premorbid IQ did not reduce rates of abnormal performance. However, it eliminated the contribution of these variables to rates of abnormal test performance. These findings raise fundamental questions about the nature and interpretation of abnormal test performance by normal, healthy adults. (JINS, 2008, 14, 436–445.)

We give a simple proof for the independence of record indices in Nevzorov's record model which is based on ranks. An application of these results to a probabilistic analysis of a particular searching algorithm with non-equiprobable orderings is also discussed.

Let X1X2, · ·· be a Markov Bernoulli sequence with initial probabilities p of success and q = 1 – p of failure, and probabilities 1 – (1 – π) p, (1 – π) p in the first row and (1 – π) (1 – p), (1 – π) p + πin the second row of the transition matrix. If we define Sn = Σi=1nXi, then the limit distribution P{Sn = k} is obtained when n →∞, np →λ.

Assuming that the original observation from a discrete distribution is subject to damage according to a binomial probability law, it is shown here that a class of discrete distributions consisting of binomial, Poisson and mixed Poisson distributions, is characterized by the generalized Rao–Rubin condition 0 <π, π′ < 1 are constants. Y denotes the resulting or observed random variable, taking the values r = 0, 1, 2, ….