In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn) evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space 𝒴 The process (Yn, Zn) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn)-conditioned process (Zn) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

A linear DNA molecule may be labelled at a fixed locus by a minute complementary radioactive molecule. A collection of identical molecules is to be so labelled and each one independently cut at a random number, N, of random places, X (N Poisson, X uniform). Fragments containing the label are to be collected and assayed by length. It is shown that the recovery pattern (fragment length distribution) contains a jump discontinuity at the fixed locus and may be used to determine the distance between the attachment site and the nearest end of the molecule.

The recovery pattern under the hypothesis that the collection of molecules are circularly permuted, i.e. the labelled locus is uniformly distributed over the length of the molecule, does not contain such discontinuities. The case where the labelling molecule has a non-negligible extent is also treated.

Characteristics of some collapsing supercritical stochastic population processes where an embedded Galton–Watson process can be isolated are shown to be identical to those of some associated subcritical process.

There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton–Watson process. Some continuity properties of the distributions of their limits are discussed.

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.

Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

Given two independent Poisson processes 𝒜 and ℬ with rates respectively α and β (α < β) we match each point of 𝒜 with the closest point of ℬ that has not already been matched. The points of 𝒜 are taken in random order. It is shown that the point process of unmatched points of ℬ is a renewal process with the same interval distribution as the busy period of an M/M/1 queue. The distribution and moments of the distance between a typical point of 𝒜 and the corresponding matched point of ℬ are obtained. Variants of the matching process in which the assigned point of ℬ must lie to the right of the point of 𝒜, and in which the matching distance must be less than a fixed tolerance are studied. The use of matched samples to control for bias in observational studies is discussed.

Let Y1, Y2, · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the Xil…im.

We consider the situation in which the decision-maker is allowed to have two choices and he must choose both the best and the second best from a group of N applicants. The optimal stopping rule and the maximum probability of choosing both of them are derived.

An expression for the characteristic function of a stationary filtered renewal process with a two-sided impact function is derived, and formulae for the moments are obtained.

In a previous paper, the author has described a method for obtaining envelope processes for oscillatory stochastic processes. These are processes which can be represented as the output of a time-varying linear filter whose input is a stationary process.

It is shown in this paper that the proposed definition of the envelope process may not be unique, but may depend on the particular representation of the oscillatory process chosen.

It is then shown that for a class of oscillatory processes which is of particular interest, the class of transient processes, there is a class of natural representations which all lead to a unique envelope process.

A matrix game is played repeatedly, with the actions taken at each stage determining both a reward paid to Player I and the probability of continuing to the next stage. An infinite history of play determines a sequence (Rn) of such rewards, to which we assign the payoff lim supn (R1 + · ·· + Rn). Using the concept of playable strategies, we slightly generalize the definition of the value of a game. Then we find sufficient conditions for the existence of a value and for the existence of stationary optimal strategies (Theorems 8 and 9). An example shows that the game need not have a value (Example 4).

Stoppable families of alternative bandit processes are decision processes with the property that at each decision epoch the choice is between allocating service to one of the constituent bandit processes or stopping and deciding in favour of one of them. The problem is considered of finding optimal (or good suboptimal) strategies for such processes. The theory for non-stoppable families leads us to study the performance of a simple strategy. This is shown to be optimal under certain conditions. These conditions are discussed and an example relating to research planning is given.

We consider the optimal wagers to be made by a gambler facing cointossing games who desires to maximize the expected value of the utility of his final fortune in a fixed number n of plays. In the case of fixed probability of a win, the optimal bet is shown to be increasing in the probability. In the case of unknown probability of a win, the wager is shown to be monotone in the prior distribution under the monotone likelihood ratio ordering of these distributions.

We investigate conditions under which the transition probabilities of various Markovian storage processes approach a stationary limiting distribution π at an exponential rate. The models considered include the waiting time of the M/G/1 queue, and models for dams with additive input and state-dependent release rule satisfying a ‘negative mean drift' condition. A typical result is that this exponential ergodicity holds provided the input process is ‘exponentially bounded'; for example, in the case of a compound Poisson input, a sufficient condition is an exponential bound on the tail of the input size distribution. The results are proved by comparing the discrete-time skeletons of the Markov process with the behaviour of a random walk, and then showing that the continuous process inherits the exponential ergodicity of any of its skeletons.

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.

The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

For a supercritical branching process x = {xn; n ≧ 0, x0 = 1} with random environments, define when xn > 0; and = 1 when xn = 0. When x is assumed to satisfy the standard regularity assumptions, under the non-extinction hypothesis, is a strongly consistent and asymptotically unbiased estimator for the criticality parameter π and is asymptotically normal. A strongly consistent estimator, is also proposed for the associated variance, σ2.

In this note we are concerned with the rate of extinction of certain continuous-time birth-death processes on the positive integers with absorption at 0. The class we deal with includes birth-death processes with mean holding time h(i) at i such that h (i)∼ i–α as i →∞, 0 ≦ α< 1. In general, our result estimates to within a constant multiple the probability of non-extinction by time t. For h(i)∼ i–α, the result states that the probability of non-extinction is of order t−1/(2-α) We give an application to interacting particle systems.

Some characterizations of continuous and discrete exponential families of distributions are given. These characterizations are used to obtain some limit theorems.

This paper studies a generalization of the GI/G/1 queueing system in which the inter-arrival times are not necessarily identically distributed and there is a random set-up time for customers who arrive when the server is idle. A recursive scheme is derived to obtain the distribution of the actual waiting time of each customer in the system.