Let Yk(ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z(ω) := ∑k≥0Yk(ω) is the total progeny of ω. In this paper, we will prove various statistical properties of Z and Yk. 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 Yk(ω) := ∑j=0kYj(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 Ykk, and show that, as n → ∞, Yk/nk is asymptotically Gaussian under the conditional distribution P(· | Z = n).

Consider a sequence (Xn) of independent and identically distributed random variables taking nonnegative integer values, and call Xn a record if Xn> max{X1,…,Xn−1}. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.

Let {Xk, k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {wk, k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1nwkXk and the maximum of weighted sums max1≤m≤n∑k=1mwkXk, subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss Xk during the kth year, with a discount or inflation factor wk.

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

The first purpose of this article is to study the phase transitions of a new interacting particle system. We consider two types of particle, each of which gives birth to particles of the same type as the parent. Particles of the second type can die, whereas those of the first type mutate into the second type. We prove that the three possible outcomes of the process, that is, extinction, survival of the type-2s, or coexistence, may each occur, depending on the selected parameters. Our second, and main, objective, however, is to investigate the duality properties of the process; the corresponding dual process exhibits a structure somewhat different to that of well-known particle systems.

We study the following model for a phylogenetic tree on n extant species: the origin of the clade is a random time in the past whose (improper) distribution is uniform on (0,∞); thereafter, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on there being the prescribed number n of species at the present time. We study various mathematical properties of this model as n→∞: namely the time of origin and of the most recent common ancestor, the pattern of divergence times within lineage trees, the time series of the number of species, the total number of extinct species, the total number of species ancestral to the extant ones, and the ‘local’ structure of the tree itself. We emphasize several mathematical techniques: the association of walks with trees; a point process representation of lineage trees; and Brownian limits.

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 n3/2, in contrast to both the subcritical case, in which it is n1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

In this study, we characterize the equilibrium behavior of spatial migration processes that represent population migrations, or birth-death processes, in general spaces. These processes are reversible Markov jump processes on measure spaces. As a precursor, we present fundamental properties of reversible Markov jump processes on general spaces. A major result is a canonical formula for the stationary distribution of a reversible process. This involves the characterization of two-way communication in transitions, using certain Radon-Nikodým derivatives. Other results concern a Kolmogorov criterion for reversibility, time reversibility, and several methods of constructing or identifying reversible processes.

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.

Let {νε, ε>0} be a family of probabilities for which the decay is governed by a large deviation principle, and consider the simulation of νε0(A) for some fixed measurable set A and some ε0>0. We investigate the circumstances under which an exponentially twisted importance sampling distribution yields an asymptotically efficient estimator. Varadhan's lemma yields necessary and sufficient conditions, and these are shown to improve on certain conditions of Sadowsky. This is illustrated by an example to which Sadowsky's conditions do not apply, yet for which an efficient twist exists.

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a Kn distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

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 (s1,s2,…) is sβφ(s1/s,s2/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.