Journal of Applied Probability

A general class of Markovian non-Gaussian bifurcating models for cell lineage data is presented. Examples include bifurcating autoregression, random coefficient autoregression, bivariate exponential, bivariate gamma, and bivariate Poisson models. Quasi-likelihood estimation for the model parameters and large-sample properties of the estimates are discussed.

The concept of complete mixability is relevant to some problems of optimal couplings with important applications in quantitative risk management. In this paper we prove new properties of the set of completely mixable distributions, including a completeness and a decomposition theorem. We also show that distributions with a concave density and radially symmetric distributions are completely mixable.

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t₁, t₂ ɛ T, t₁ 〈 t₂; the random variable X(t₂) – X(t₁) is called the increment of the process X(t) over the interval [t₁, t₂]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plim_τ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, lim_τ→0P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

A simple Bayesian model for oil exploration is suggested to investigate strategies for drilling. A condition on the way successes and failures affect the prior distribution implies a certain form of the detection mechanism. It is shown that the problem of finding strategies for drilling reduces to an optimal stopping problem. Two new families of distributions are obtained with generating functions related to classical work on partitions of integers. By using such distributions and simple mixtures of them as priors, the stopping problem can be solved explicitly. This leads to the construction of simple strategies and their effectiveness is demonstrated by evaluating suitable operating characteristics.

The paper re-examines Quinn and MacGillivray's (1986) stationary birth-death process for a population of fixed size N consisting of two types of parasite, active and passive, and sets up a more elaborate model for the dichotomy between parasites on hosts with and without open wounds resulting from previous parasite attacks. The probability generating functions for the stationary count distributions are obtained, allowing limiting forms of the distributions to be investigated.

The paper puts forward steady-state Markov chain models for the Heine and Euler distributions. The models for oil exploration strategies that were discussed by Benkherouf and Bather (1988) are reinterpreted as current-age models for discrete renewal processes. Steady-state success-runs processes with non-zero probabilities that a trial is abandoned, Foster processes, and equilibrium random walks corresponding to elective M/M/1 queues are also examined.

The inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution. The probabilities for the direct absorption distribution are obtained via the inverse absorption probabilities and exact expressions for its first two factorial moments are derived using q-series transformations of its probability generating function. Alternative models for the distribution are given.

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.

A definition of ESS (evolutionarily stable strategy) is suggested for games in which there are two types of player, each with its own set of strategies, and the fitness of any strategy depends upon the strategy mix, of both types, in the population as a whole. We check that the standard ESS results hold for this definition and discuss a mate-desertion model which has appeared in the literature in which the two types are male and female.

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the formf(n) =c(m+n)^ß, where m is an arbitrary non-negative parameter andcis not 0. For – ½ <ß< 0 the Hurst exponent is shown to be precisely given by 1 +ß.Forß≦ – ½ and forß= 0 the Hurst exponent is 0.5, while forß > 0it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.