A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj <∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.

We study a problem when the optimal stopping for a one-dimensional diffusion process is generated by a threshold strategy. Namely, we give necessary and sufficient conditions (on the diffusion process and the payoff function) under which a stopping set has a threshold structure.

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 consider degenerate limit laws for the sequence {Xn, n}n(N of successive maxima of identically distributed random variables. It turns out that the concentration of Xn, n for large n can be determined in terms of a tail ratio of the underlying distribution function F. Applications to the outlier-behaviour of probability distributions are given.

Conditions that have been widely used to establish stochastic ordering of random vectors include those given by Kamae et al. (1977), Barlow and Proschan (1975) and Langberg (1988). In this note we offer conditions that are weaker than those above, thus providing potential application to a wider class of problems.

In recent publications of Hadwiger, Bokowski, and Wills it has been shown that it is possible to evaluate the average distance between a fixed and a randomly distributed convex set in terms of the mean projection measures of these sets. In the present note it is pointed out that these and more general results can be obtained from a theorem concerning the relationship between integrals over arbitrary measure spaces and corresponding Lebesgue–Stieltjes integrals.

We consider a bivariate Markov counting process with transition probabilities having a particular structure, which includes a number of useful population processes. Using a suitable random time-scale transformation, we derive some probability statements about the process and some asymptotic results. These asymptotic results are also derived using martingale methods. Further, it is shown that these methods and results can be used for inference on the rate parameters for the process. The general epidemic model and the square law conflict model are used as illustrative examples.

An optimal allocation of subsystems depending on the system structure and reliability ordering of inherent subsystem components is determined, in the presence of various external influences on the reliability of components in different locations. It is carried out with the help of L-superadditive functions and Schur-convex functions.

Stochastic optimization of semimartingales which permit a dynamic description, like a stochastic differential equation, leads normally to dynamic programming procedures. The resulting Bellman equation is often of a very genera! nature, and analytically hard to solve. The models in the present paper are formulated in terms of the relative change, and the optimality criterion is to maximize the expected rate of growth. We show how this can be done in a simple way, where we avoid using the Bellman equation. An application is indicated.

This paper examines the infinitely high dam with seasonal (periodic) Lévy input under the unit release rule. We show that a periodic limiting distribution of dam content exists whenever the mean input over a seasonal cycle is less than 1. The Laplace transform of dam content at a finite time and the Laplace transform of the periodic limiting distribution are derived in terms of the probability of an empty dam. Necessary and sufficient conditions for the periodic limiting distribution to have finite moments are given. Convergence rates to the periodic limiting distribution are obtained from the moment results. Our methods of analysis lean heavily on the coupling method and a stochastic monotonicity result.

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number cd > 0 almost surely as the number of steps goes to ∞, and cd ~ (eπ)1/2d1/4e-2√d holds as d → ∞.

In an optimal variance stopping problem the goal is to determine the stopping time at which the variance of a sequentially observed stochastic process is maximized. A solution method for such a problem has been recently provided by Pedersen (2011). Using the methodology developed by Pedersen and Peskir (2012), our aim is to show that the solution to the initial problem can be equivalently obtained by constraining the variance stopping problem to the expected size of the stopped process and then by maximizing the solution to the latter problem over all the admissible constraints. An application to a diffusion process used for modeling the dynamics of interest rates illustrates the proposed technique.

In this note, we consider discrete-time finite Markov chains and assume that they are only partly observed. We obtain finite-dimensional normalized filters for basic statistics associated with such processes. Recursive equations for these filters are derived by means of simple computations involving conditional expectations. An application to the estimation of parameters of the so-called discrete-time batch Markovian arrival process is outlined.

The recurrent epidemic model has hitherto proved mathematically intractable. It is shown here that by imposing the mild condition that the source of new susceptibles is initially of a constant size M for M sufficiently large that there is zero probability of more than M new susceptibles being added to the population in the course of the epidemic, then state probabilities for the stochastic recurrent epidemic model can be found.

We call `bits' a sequence of devices indexed by positive integers, where every device can be in two states: 0 (idle) and 1 (active). Start from the `ground state' of the system when all bits are in 0-state. In our first binary flipping (BF) model the evolution of the system behaves as follows. At each time step choose one bit from a given distribution P on the positive integers independently of anything else, then flip the state of this bit to the opposite state. In our second damaged bits (DB) model a `damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both the BF and DB models to show recurrent or transient behaviour, depending on the properties of the distribution P. We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a central limit theorem for the number of active bits for both models.

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.

A population of susceptible individuals partitioned into m groups and exposed to a contagious disease is considered.

The progress of this simple epidemic is modeled by an m -dimensional stochastic process, whose components are the number of infective individuals in the respective groups at time t. Exact and approximate formulas for the joint and marginal state probabilities are obtained.

Gorostiza and Wakolbinger (1991), and Dawson and Perkins (1991) established the same persistence criterion for a class of critical branching particle systems and for a class of superprocesses respectively. In this note we take another approach to the criterion and present a simpler proof of it.

We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.

The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general d-dimensional case is also established.