For n independent, identically distributed uniform points in [0, 1]d, d ≥ 2, let Ln be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish the asymptotics of the mean and the variance of Ln, and show that Ln satisfies a central limit theorem, unlike in the case d = 2.

In this paper, we study the extremal behavior of stationary mixed moving average processes of the form Y(t)=∫ℝ+×ℝf(r,t-s) dΛ(r,s), t∈ℝ, where f is a deterministic function and Λ is an infinitely divisible, independently scattered random measure whose underlying driving Lévy process is regularly varying. We give sufficient conditions for the stationarity of Y and compute the tail behavior of certain functionals of Y. The extremal behavior is modeled by marked point processes on a discrete-time skeleton chosen properly by the jump times of the underlying driving Lévy process and the extremes of the kernel function. The sequences of marked point processes converge weakly to a cluster Poisson random measure and reflect extremes of Y at a high level. We also show convergence of the partial maxima to the Fréchet distribution. Our models and results cover short- and long-range dependence regimes.

We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k∈ℕ is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

We derive a large deviation principle for a Brownian immigration branching particle system, where the immigration is governed by a Poisson random measure with a Lebesgue intensity measure.

Let denote the class of local subexponential distributions and F∗ν the ν-fold convolution of distribution F, where ν belongs to one of the following three cases: ν is a random variable taking only a finite number of values, in particular ν ≡ n for some n ≥ 2; ν is a Poisson random variable; or ν is a geometric random variable. Along the lines of Embrechts, Goldie, and Veraverbeke (1979), the following assertion is proved under certain conditions: This result is applied to the infinitely divisible laws and some new results are established. The results obtained extend the corresponding findings of Asmussen, Foss, and Korshunov (2003).

Given an ℝd-valued supercritical branching Wiener process, let ψ(A, T) be the number of particles in A ⊂ ℝd at time T (T = 0, 1, 2, …). We provide a complete asymptotic expansion of ψ(A, T) as T → ∞, generalizing the work of X. Chen.

In this paper, we propose a new urn model. A single urn contains b black balls and w white balls. For each observation, we randomly draw m balls and note their colors, say k black balls and m − k white balls. We return the drawn balls to the urn with an additional ck black balls and c(m − k) white balls. We repeat this procedure n times and denote by Xn the fraction of black balls after the nth draw. To investigate the asymptotic properties of Xn, we first perform some computational studies. We then show that {Xn} forms a martingale, which converges almost surely to a random variable X. The distribution of X is then shown to be absolutely continuous.

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅,⋅) and whose cumulative process satisfies the LDP with rate function I(⋅). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I.

Let X1, X2,… be real-valued random variables. For u>0, define the time of ruin T = T(u) by T = inf{n: X1+⋯+Xn>u} or T=∞ if X1+⋯+Xn≤u for every n = 1,2,…. We are interested in the ruin probabilities of general processes {Xn} for large u. In the presence of heavy tails, one often finds power estimates. Our objective is to specify the associated powers and provide the crude estimate P(T≤xu)≈u−R(x) for large u, for a given x∈ℝ. The rate R(x) will be described by means of tails of partial sums and maxima of {Xn}. We also extend our results to the case of the infinite time horizon.

Consider a random walk S=(Sn: n≥0) that is ‘perturbed’ by a stationary sequence (ξn: n≥0) to produce the process S=(Sn+ξn: n≥0). In this paper, we are concerned with developing limit theorems and approximations for the distribution of Mn=max{Sk+ξk: 0≤k≤n} when the random walk has a drift close to 0. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walks and extreme-value limit theory.

Berry-Esseen-type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated in an application to combinatorial central limit theorems in which the random permutation has either the uniform distribution or one that is constant over permutations with the same cycle type, with no fixed points. The size biasing bounds are applied to the occurrences of fixed, relatively ordered subsequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs in finite graphs.

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

We consider a multi-type branching random walk on d-dimensional Euclidian space. The~uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) ‘Perron-Frobenius theory’ for matrices that are smooth functions of a variable λ∈L and are nonnegative when λ∈L−⊂L, where L is an open set in ℂd, and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

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.

A random variable Y is branching stable (B-stable) for a nonnegative integer-valued random variable J with E(J)>1 if Y*J∿cY for some scalar c, where Y*J is the sum of J independent copies of Y. We explore some aspects of this notion of stability and show that, for any Y0 with finite nonzero mean, if we define Yn+1=Yn*J/E(J) then the sequence Yn converges in law to a random variable Y∞ that is B-stable for J. Also Y∞ is the unique B-stable law with mean E(Y0). We also present results relating to random variables Y0 with zero means and infinite means. The notion of B-stability arose in a scheme for cataloguing a large network of computers.

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

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.

We use a sample-path technique to derive asymptotics of generalized Jackson queueing networks in the fluid scale; that is, when space and time are scaled by the same factor n. The analysis only presupposes the existence of long-run averages and is based on some monotonicity and concavity arguments for the fluid processes. The results provide a functional strong law of large numbers for stochastic Jackson queueing networks, since they apply to their sample paths with probability 1. The fluid processes are shown to be piecewise linear and an explicit formulation of the different drifts is computed. A few applications of this fluid limit are given. In particular, a new computation of the constant that appears in the stability condition for such networks is given. In a certain context of a rare event, the fluid limit of the network is also derived explicitly.

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.