This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.

For a stationary long-range dependent point process N(.) with Palm distribution P0, the Hurst index H ≡ sup{h : lim sup t→∞t-2h var N(0,t] = ∞} is related to the moment index κ ≡ sup{k : E0(Tk) < ∞} of a generic stationary interval T between points (E0 denotes expectation with respect to P0) by 2H + κ ≥ 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which κ < 2 is necessarily long-range dependent with Hurst index greater than ½. An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., E0(T2) < ∞.

A manpower system of constant size, controlled by recruitment, is described by a partially stochastic model in which there are fixed promotion rates, no demotions and stochastic wastage. The geometric-probabilistic relationship is examined for the attainability after one step and after two steps.

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1, b)-coalescent by exploiting the method of sequential approximations.

We study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function, which is a function of its weight and degree, and connects to $\ell$ new-coming vertices. Under a certain technical assumption, applying the theory of Crump–Mode–Jagers branching processes, we derive formulas for the limiting distributions of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we rigorously prove observations of Bianconi related to the evolving Cayley tree (Phys. Rev. E 66, paper no. 036116, 2002). We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call ‘generalised preferential attachment with fitness’. We show that this model can exhibit condensation, where a positive proportion of edges accumulates around vertices with maximal weight, or, more drastically, can have a degenerate limiting degree distribution, where the entire proportion of edges accumulates around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process.

In this paper we investigate the ‘local’ properties of a random mapping model, TnD̂, which maps the set {1, 2, …, n} into itself. The random mapping TnD̂, which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a collection of exchangeable random variables D̂1, …, D̂n which satisfy In the random digraph, GnD̂, which represents the mapping TnD̂, the in-degree sequence for the vertices is given by the variables D̂1, D̂2, …, D̂n, and, in some sense, GnD̂ can be viewed as an analogue of the general independent degree models from random graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of GnD̂ - for example, the numbers of predecessors and successors of v in GnD̂. We show that the distribution of several variables associated with the local structure of GnD̂ can be expressed in terms of expectations of simple functions of D̂1, D̂2, …, D̂n. We also consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest.

In this paper we investigate the asymptotic behavior of sequences of successive Steiner and Minkowski symmetrizations. We state an equivalence result between the convergences of those sequences for Minkowski and Steiner symmetrizations. Moreover, in the case of independent (and not necessarily identically distributed) directions, we prove the almost-sure convergence of successive symmetrizations at exponential rate for Minkowski, and at rate with c > 0 for Steiner.

We consider Monte Carlo methods for the classical nonlinear filtering problem. The first method is based on a backward pathwise filtering equation and the second method is related to a backward linear stochastic partial differential equation. We study convergence of the proposed numerical algorithms. The considered methods have such advantages as a capability in principle to solve filtering problems of large dimensionality, reliable error control, and recurrency. Their efficiency is achieved due to the numerical procedures which use effective numerical schemes and variance reduction techniques. The results obtained are supported by numerical experiments.

We model and study the genetic evolution and conservation of a population of diploid hermaphroditic organisms, evolving continuously in time and subject to resource competition. In the absence of mutations, the population follows a three-type, nonlinear birth-and-death process, in which birth rates are designed to integrate Mendelian reproduction. We are interested in the long-term genetic behavior of the population (adaptive dynamics), and in particular we compute the fixation probability of a slightly nonneutral allele in the absence of mutations, which involves finding the unique subpolynomial solution of a nonlinear three-dimensional recurrence relationship. This equation is simplified to a one-dimensional relationship which is proved to admit exactly one bounded solution. Adding rare mutations and rescaling time, we study the successive mutation fixations in the population, which are given by the jumps of a limiting Markov process on the genotypes space. At this time scale, we prove that the fixation rate of deleterious mutations increases with the number of already fixed mutations, which creates a vicious circle called the extinction vortex.

The aim of this note is to prove the following characterization.

In this paper, isotropic random projections of d-sets in ℝn are studied, where a d-set is a subset of a d-dimensional affine subspace which satisfies certain regularity conditions. The squared volume reduction induced by the projection of a d-set onto an isotropic random p-subspace is shown to be distributed as a product of independent beta-distributed random variables, for d ≤ p. One of the proofs of this result uses Wilks' lambda distribution from multivariate normal theory. The result is related to Cauchy's and Crofton's formulae in stochastic geometry. In particular, it can be used to give a new and quite simple proof of one of the classical Crofton intersection formulae.

This paper analyzes optimal single and multiple stopping rules for a class of correlated random walks that provides an elementary model for processes exhibiting momentum or directional reinforcement behavior. Explicit descriptions of optimal stopping rules are given in several interesting special cases with and without transaction costs. Numerical examples are presented comparing optimal strategies to simpler buy and hold strategies.

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, rn, f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D1nrn2. In the second part of the paper we consider the contact process ξt on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c1 and c2 such that, with probability at least 1 - c1 / n4, the contact process starting with all nodes infected survives up to time tn = exp(c2n/logn) for all n.

It is shown that no condition of the form E[g(N)] =∞, where N denotes a typical family size, is sufficient for explosiveness of a Markov branching process.

We consider two variants of M/G/1 queues with exhaustive service and multiple vacations; (1) customers cannot wait for their services longer than an interval of length T, and (2) customers cannot stay in the system longer than an interval of length T. We show that the probability distribution functions of the waiting times for the two systems are given in terms of those for the corresponding M/G/1 vacation systems without any residence-time limits.

We consider partial customer flexibility in service systems under two different designs. In the first design, flexible customers have their own queue and each server has its own queue of dedicated customers. Under this model, the problem is a scheduling problem and we show under various settings that the dedicated customers first (DCF) policy is optimal. In the second design, flexible customers are not queued separately and must be routed to one of the server's dedicated queues upon arrival. We extend earlier results about the ‘join the smallest work (JSW)’ policy to systems with dedicated as well as flexible arrivals. We compare these models to a routeing model in which only the queue length is available in terms of both efficiency and fairness and argue that the overall best approach for call centers is JSW routeing. We also discuss how this can be implemented in call centers even when work is unknown.

In this paper we consider the observed lifetime of a multistate monotone system and ‘the critical lower set' which causes the system deterioration. Then, under suitable conditions, we identify the component lifetime distribution using a Newton–Kantorovic iterative procedure as in Meilijson (1981).

We define the ‘linear scan transform' G of a set in ℝd using information observable on its one-dimensional linear transects. This transform determines the set covariance function, interpoint distance distribution, and (for convex sets) the chord length distribution. Many basic integral-geometric formulae used in stereology can be expressed as identities for G. We modify a construction of Waksman (1987) to construct a metric η for ‘regular' subsets of ℝd defined as the L1 distance between their linear scan transforms. For convex sets only, η is topologically equivalent to the Hausdorff metric. The set covariance function (of a generally non-convex set) depends continuously on its set argument, with respect to η and the uniform metric on covariance functions.

We develop a new, self-contained proof that the expected number of generations required for gene allele fixation or extinction in a population of size n is O(n) under general assumptions. The proof relies on a discrete Markov chain analysis. We further develop an algorithm to compute expected fixation or extinction time to any desired precision. Our proofs establish O(nH(p)) as the expected time for gene allele fixation or extinction for the Fisher-Wright problem, where the gene occurs with initial frequency p and H(p) is the entropy function. Under a weaker hypothesis on the variance, the expected time is O(n(p(1-p))1/2) for fixation or extinction. Thus, the expected-time bound of O(n) for fixation or extinction holds in a wide range of situations. In the multi-allele case, the expected time for allele fixation or extinction in a population of size n with n distinct alleles is shown to be O(n). From this, a new proof is given of a coalescence theorem about the mean time to the most recent common ancestor (MRCA), which applies to a broad range of reproduction models satisfying our mean and weak variation conditions.