We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2, in contrast to both the subcritical case, in which it is n 1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

We generalize a theorem due to Keilson on the approximate exponentiality of waiting times for rare events in regenerative processes. We use the result to investigate the limit distribution for a family of first entrance times in a sequence of Ehrenfest urn models. As a second application, we consider approximate pattern matching, a problem arising in molecular biology and other areas.

Diffusion models in economics, finance, queueing, mathematical biology, and electrical engineering often involve reflecting barriers. In this paper, we study the analytical representation of transition densities for reflected one-dimensional diffusions in terms of their associated Sturm-Liouville spectral expansions. In particular, we provide explicit analytical expressions for transition densities of Brownian motion with drift, the Ornstein-Uhlenbeck process, and affine (square-root) diffusion with one or two reflecting barriers. The results are easily implementable on a personal computer and should prove useful in applications.

In this paper, we consider a certain type of space- and time-fractional kinetic equation with Gaussian or infinitely divisible noise input. The solutions to the equation are provided in the cases of both bounded and unbounded domains, in conjunction with bounds for the variances of the increments. The role of each of the parameters in the equation is investigated with respect to second- and higher-order properties. In particular, it is shown that long-range dependence may arise in the temporal solution under certain conditions on the spatial operators.

In this paper, we investigate the asymptotic behaviour of controlled branching processes with random control functions. In a critical case, we establish sufficient conditions for both their almost-sure extinction and for their nonextinction with a positive probability. For some suitably chosen norming constants, we also determine different kinds of limiting behaviour for this class of processes.

Motivated by a problem arising in the mining industry, we estimate the energy ε(η) that is needed to reduce a unit mass to fragments of size at most η in a fragmentation process, when η→0. We assume that the energy used in the instantaneous dislocation of a block of size s into a set of fragments (s 1,s 2,…) is s βφ(s 1/s,s 2/s,…), where φ is some cost function and β a positive parameter. Roughly, our main result shows that if α>0 is the Malthusian parameter of an underlying Crump-Mode-Jagers branching process (with α = 1 when the fragmentation is mass-conservative), then there exists a c∈(0,∞) such that ε(η)∼cηβ-α when β<α. We also obtain a limit theorem for the empirical distribution of fragments of size less than η that result from the process. In the discrete setting, the approach relies on results of Nerman for general branching processes; the continuous approach follows by considering discrete skeletons. In the continuous setting, we also provide a direct approach that circumvents restrictions induced by the discretization.

A multitype branching process is presented in the framework of marked trees and its structure is studied by applying the strong branching property. In particular, the Markov property and the expression for the generator are derived for the process whose components are the numbers of particles of each type. The filtering of the whole population, observing the number of particles of a given type, is discussed. Weak uniqueness for the filtering equation and a recursive structure for the linearized filtering equation are proved under a suitable assumption on the reproduction law.

In this paper, we provide a novel approach to studying the heavy-tailed asymptotics of the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed from two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence, respectively. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy tailed. Then we derive the long-tailed asymptotics of the R-measure in terms of the RG-factorization of the repeating matrix sequence, and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis of the subexponential asymptotics of the stationary probability vector.

We consider a heterogeneous population of identical particles divided into a finite number of classes according to their level of health. The partition can change over time, and a suitable exchangeability assumption is made to allow for having identical items of different types. The partition is not observed; we only observe the cardinality of a particular class. We discuss the problem of finding the conditional distribution of particle lifetimes, given such observations, using stochastic filtering techniques. In particular, a discrete-time approximation is given.

Consider a continuous-time Markov chain evolving in a random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included.

In this paper, we consider a risk model in which each main claim induces a delayed claim called a by-claim. The time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. From martingale theory, an expression for the ultimate ruin probability can be derived using the Lundberg exponent of the associated nondelayed risk model. It can be shown that the Lundberg exponent of the proposed risk model is the same as that of the nondelayed one. Brownian motion approximations for ruin probabilities are also discussed.

We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed.

As proposed by Irle and Gani in 2001, a process X is said to be slower in level crossing than a process Y if it takes X stochastically longer to exceed any given level than it does Y. In this paper, we extend a result of Irle (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and M/M/s/c systems.

Let Y k (ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z(ω) := ∑k≥0 Y k (ω) is the total progeny of ω. In this paper, we will prove various statistical properties of Z and Y k . We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that Y k (ω) := ∑j=0 k Y j (ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree ω. We then proceed to study the joint probability distribution of Z and Y k k , and show that, as n → ∞, Y k /n k is asymptotically Gaussian under the conditional distribution P(· | Z = n).

In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.

In practical situations, we observe the number of claims to an insurance portfolio but not the claim intensity. It is therefore of interest to try to solve the ‘filtering problem’; that is, to obtain the best estimate of the claim intensity on the basis of reported claims. In order to use the Kalman-Bucy filter, based on the Cox process incorporating a shot noise process as claim intensity, we need to approximate it by a Gaussian process. We demonstrate that, if the primary-event arrival rate of the shot noise process is reasonably large, we can then approximate the intensity, claim arrival, and aggregate loss processes by a three-dimensional Gaussian process. We establish weak-convergence results. We then use the Kalman-Bucy filter and we obtain the price of reinsurance contracts involving high-frequency events.

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R + n that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

We consider two versions of a simple evolutionary algorithm (EA) model for protein folding at zero temperature, namely the (1 + 1)-EA on the LeadingOnes problem. In this schematic model, the structure of the protein, which is encoded as a bit-string of length n, is evolved to its native conformation through a stochastic pathway of sequential contact bindings. We study the asymptotic behavior of the hitting time, in the mean case scenario, under two different mutations: the one-flip, which flips a unique bit chosen uniformly at random in the bit-string, and the Bernoulli-flip, which flips each bit in the bit-string independently with probability c/n, for some c ∈ ℝ + (0 ≤ c ≤ n). For each algorithm, we prove a law of large numbers, a central limit theorem, and compare the performance of the two models.

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.