We study discrete-time population models where the nearest future of an individual may depend on the individual's life-stage (age and reproduction history) and the current population size. A criterion is given for whether there is a positive probability that the population survives forever. We identify the cases when population size grows exponentially and linearly and show that in the latter population size scaled by time is asymptotically Γ-distributed.

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.

The paper concerns the asymptotic distributions of cluster functionals of extreme events in a dth-order stationary Markov chain {Xn, n = 1,2,…} for which the joint distribution of (X1,…,Xd+1) is absolutely continuous. Under some distributional assumptions for {Xn}, we establish weak convergence for a class of cluster functionals and obtain representations for the asymptotic distributions which are well suited for simulation. A number of examples important in applications are presented to demonstrate the usefulness of the results.

This work presents an estimate of the error on a cumulative reward function until the entrance time of a continuous-time Markov chain into a set, when the infinitesimal generator of this chain is perturbed. The derivation of an error bound constitutes the first part of the paper while the second part deals with an application where the time until saturation is considered for a circuit switched network which starts from an empty state and which is also subject to possible failures.

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

This paper arose from interest in assessing the quality of random number generators. The problem of testing randomness of a string of binary bits produced by such a generator gained importance with the wide use of public key cryptography and the need for secure encryption algorithms. All such algorithms are based on a generator of (pseudo) random numbers; the testing of such generators for randomness became crucial for the communications industry where digital signatures and key management are vital for information processing.

The concept of approximate entropy has been introduced in a series of papers by S. Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is based on incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness, in a sequence. Pincus and Kalman (1997) evaluated approximate entropies for binary and decimal expansions of e, π, √2 and √3 with the surprising conclusion that the expansion of √3 demonstrated much less irregularity than that of π. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomness based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now this distribution remained unknown and was thought to be difficult to obtain. To derive the limiting distribution of approximate entropy we modify its definition. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ2-random variable. The P-values of approximate entropy test statistics for binary expansions of e, π and √3 are plotted. Although some of these values for √3 digits are small, they do not provide enough statistical significance against the randomness hypothesis.

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

The estimation of critical values is one of the most interesting problems in the study of interacting particle systems. The bounds obtained analytically are not usually very tight and, therefore, computer simulation has been proved to be very useful in the estimation of these values. In this paper we present a new method for the estimation of critical values in any interacting particle system with an absorbing state. The method, based on the asymptotic behaviour of the absorption time of the process, is very easy to implement and provides good estimates. It can also be applied to processes different from particle systems.

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.

By applying a perturbation method to the structured coalescent process in population genetics theory, we obtain the approximate solutions of the moment generating functions for the total coalescence time, the number of segregating sites among sampled DNA sequences and the number of allele types in a sample in the case of strong migration.

We study a system of interacting renewal processes which is a model for neuronal activity. We show that the system possesses an exponentially large number (with respect to the number of neurons in the network) of limiting configurations of the ‘firing neurons’. These we call patterns. Furthermore, under certain conditions of symmetry we find an algorithm to control limiting patterns by means of the connection parameters.

Minimal repairs have been given considerable attention in the reliability literature. Instead of replacing a failed system by a new one, such a minimal repair restores the system to the state it had just before failure. But the state just before failure depends on the information which is available about the system. Different information levels are possible. This paper gives a general definition characterizing point processes which describe time points of minimal repairs with respect to a certain information level. Some examples demonstrate the wide range of applications.

We consider a network with N infinite-buffer queues with service rates λ, and global task arrival rate Nν. Each task is allocated L queues among N with uniform probability and joins the least loaded one, ties being resolved uniformly. We prove Q-chaoticity on path space for chaotic initial conditions and in equilibrium: any fixed finite subnetwork behaves in the limit N goes to infinity as an i.i.d. system of queues of law Q. The law Q is characterized as the unique solution for a non-linear martingale problem; if the initial conditions are q-chaotic, then Q0 = q, and in equilibrium Q0 = qρ is the globally attractive stable point of the Kolmogorov equation corresponding to the martingale problem. This result is equivalent to a law of large numbers on path space with limit Q, and implies a functional law of large numbers with limit (Qt)t≥0. The significant improvement in buffer utilization, due to the resource pooling coming from the choices, is precisely quantified at the limit.

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.

Hoeffding's inequality can be used in conjunction with the declared parameters of a traffic source, such as its peak rate, to obtain confidence intervals for measurements of the traffic's effective bandwidth. We describe a variety of interval-estimation procedures based on this idea, designed to provide differing degrees of robustness against non-stationarity. We also discuss how to compute confidence intervals for the effective bandwidth of an aggregate of traffic sources.

Let Xt be a Feller (branching) diffusion with drift αx. We consider new processes, the probability measures of which are obtained from that of X via changes of measure involving suitably normalized exponential functions of with λ > 0. The new processes can be thought of as ‘self-reinforcing’ versions of the old.

Depending on the values of α, T and λ, the process under the new measure is shown to exhibit explosion in finite time. We also obtain a number of other results related to the new processes.

Since the Feller diffusion is also the total mass process of a superprocess, we relate the finite-time explosion property to the behaviour of superprocesses with local self-interaction, and raise some interesting questions for these.

Let τr denote the hitting time of B(0:r) for a multidimensional diffusion process. We give verifiable criteria for finiteness/infiniteness of As an application we exhibit classes of diffusion processes which are recurrent but is infinite for all p > 0, |x| > r > 0; this includes the two-dimensional Brownian motion and the reflecting Brownian motion in a wedge with a certain parameter α = 0.

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

In the literature on mixed Poisson processes, a number of characterisation properties have been studied. As a new characterisation property for mixed Poisson processes, we show that normalised event occurrence times are the order statistics of independent uniform random variables on (0,1). Berman's theorem on lp-isotropic sequences is applied to prove the results.