We show for a certain class of Galton–Watson branching processes in varying environments (Zn)n that moments of the maximum random variable supnZn/Cn exist if and only if the same moments of limnZn/Cn exist, where Cn is a sequence of suitable constants.

Criteria are given to determine whether a given finite Markov chain can be lumped weakly with respect to a given partition of its state space. These conditions are given in terms of solution classes of systems of linear equations associated with the transition probability matrix of the Markov chain and the given partition.

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

Moment recursion relations have previously been derived for the stationary probability density functions of continuous-time stochastic systems with Wiener (white noise) input. These results are extended in this paper to the case of Poisson (shot noise) input. The non-linear dynamical systems are expressed, in general, as stochastic differential equations, with an independent increment input. The transition probability density function evolves according to the appropriate Kolmogorov equation. Moments of the stationary density are obtained from the Fourier transform of the stationary density. The moment relations can be used to estimate the parameters of linear and non-linear stochastic systems from empirical moments, given either Wiener or Poisson input.

There are m urns each containing k cells; n balls are assigned to the m urns in such a way that each cell contains at most one ball. Let Mt be the number of urns containing exactly t balls (t = 0, 1, •••, k). In the paper, the distribution of Mt, its moments and moment-generating function are obtained. The derivation of the last two seems to be new.

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

Families of alternative bandit processes have been used as models for problems in a variety of areas. Optimal strategies for these decision processes are determined by dynamic allocation indices. These indices are here shown to play an important role in the evaluation of suboptimal strategies.

Optimal stopping rules under various conditions are obtained for a proofreader who has a probability p (known or unknown) of detecting a misprint in proofsheets which contain an unknown but Poisson-distributed number of misprints.

A number of items are arranged in a line. At each unit of time one of the items is requested, the i th being requested with probability Pi. We consider rules which reorder the items between successive requests in a fashion which depends only on the position in which the most recently requested item was found. It has been conjectured that the rule which always moves the requested item one closer to the front of the line minimizes the average position of the requested item. An example with six items shows that the conjecture is false.

Consider as a model for any serial list a bookshelf with books B1, ···, Bn. At each unit of time book is demanded with probability pi and is replaced one position to the left of where it was removed (or in the same position if it is already at the left-hand end). If this transposition is made only when Bi has been demanded k times in a row, we show that the average position of the next book demanded is a monotone decreasing function of k.

We present a problem of finding the optimum inspection procedure for a system whose time to failure is an exponential random variable. We characterize a simple and surprising solution to the problem.