It is well known that a random set determines its random coverage measure. The paper gives a necessary and sufficient condition for the reverse implication. An equivalent formulation of the condition constitutes a first step in the search for a way to recognize a random measure as being the random coverage measure of a random set.

We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form d Xt=σ(Xt,Xt)dWt, where Xt= m ∧ inf0 ≤s≤tXs. In particular, we compute the expected time for X to leave an interval, classify the boundary behavior at 0, and derive a new occupation time formula for X. We also show that (Xt,Xt) admits a joint density, which can be characterized in terms of two independent tied-down Brownian meanders (or, equivalently, two independent Bessel-3 bridges). Finally, we show that the joint density satisfies a generalized forward Kolmogorov equation in a weak sense, and we derive a new forward equation for down-and-out call options.

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

The time to extinction of a subcritical Galton–Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process.

We show for the three-node Jackson network studied in [3] that a customer’s sojourn times in nodes 1 and 3 are positively correlated. We actually prove a stronger result, that the two sojourn times are associated random variables. Our proof uses a stochastic ordering argument similar to that in [4].

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {Gi(t)}, together with a probability generating function, h(s) = Σk∞= 0pksk. An ith generation particle lives a random length of time, determined by Gi(t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution Gi + 1 (t).

Let Zi(t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Zi(t) and it is shown that for some constants Ni, γ, a, Zi (t)/(Nitγ-1eαt) converges in mean square to a proper random variable.

Both small-world models of random networks with occasional long-range connections and gossip processes with occasional long-range transmission of information have similar characteristic behaviour. The long-range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.

We consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau > 2$. For all subgraphs H, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a Pólya urn model.

In this paper we derive bounds on the conductance and hence on the spectral gap of a Metropolis algorithm with a monotone, log-concave target density on an interval of ℝ. We show that the minimal conductance set has measure ½ and we use this characterization to bound the conductance in terms of the conductance of the algorithm restricted to a smaller domain. Whereas previous work on conductance has resulted in good bounds for Markov chains on bounded domains, this is the first conductance bound applicable to unbounded domains. We then show how this result can be combined with the state-decomposition theorem of Madras and Randall (2002) to bound the spectral gap of Metropolis algorithms with target distributions with monotone, log-concave tails on ℝ.

In this paper, a very general class of branching processes on the d-dimensional square lattice is studied. It is assumed that the division rates as well as the spatial distribution of offspring are configuration-dependent. The main interest of this paper is in the asymptotic geometrical behaviour of such processes. Utilizing techniques mainly due to Richardson [28], we derive conditions which are necessary and sufficient for such branching processes to have the following property: there exists a norm N(·) on Rd such that, for all 0 < ∊ < 1, we have that almost surely for all sufficiently large t, all sites in the N-ball of radius (1 – ∊)t are contained in (the set of sites occupied at time t) and is contained in the set of all sites in the N-ball of radius (1 + ∊)t (given that the process starts with finitely many particles).

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w1), …, N(S,wd)), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O((log n) / √n) in the i.i.d. case and O((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral or, more generally, together with a finite number of iterated time integrals). We show that in this case there can be no Markovian maximal coupling. Indeed, there can be no efficient Markovian coupling strategy (efficient for all pairs of distinct starting values), where the notion of efficiency extends the terminology of Burdzy and Kendall (2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling strategy.

Stochastic majorization is a tool that has been used in many areas of probability and statistics (such as multivariate statistical analysis, queueing theory and reliability theory) in order to obtain useful bounds and inequalities. In this paper we study the relations among several notions of stochastic majorization and stochastic convexity and obtain sufficient (and sometimes necessary) conditions which imply some of these notions. Extensions and generalizations of several results in the literature are obtained. Some examples and applications regarding stochastic comparisons of order statistics are also presented in order to illustrate the results of the paper.

In Part I (Hunter) a study of feedback queueing models was initiated. For such models the queue-length process embedded at all transition points was formulated as a Markov renewal process (MRP). This led to the observation that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also MRP. Part I concentrated on the properties of the embedded discrete-time Markov chains. In this part we examine the semi-Markov processes associated with each of these embedded MRP and derive expressions for the stationary distributions associated with their irreducible subspaces. The special cases of birth-death queues with instantaneous state-dependent feedback, M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback are considered in detail. The results obtained complement those derived in Part II (Hunter) for birth-death queues without feedback.

Knowing the (geometric) covariogram of a convex body is equivalent to knowing, for each direction u, the distribution of the lengths of the chords of that body which are parallel to u. We prove that the covariogram determines convex polygons, among all convex bodies, up to translation and reflection. This gives a partial answer to a problem posed by Matheron.

Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

Let Z(t) be the population size at time t in a general age-dependent branching process (as defined by Crump and Mode, or Jagers) in which the number N of offspring of a parent has expected value 1 (critical case). Assuming positivity and finiteness of the second moments of N, of the lifespan distribution and of the expected number of births per parent as a function of age (also assumed to be strongly non-lattice), the distribution of Z(t)/t conditioned on non-extinction at time t is asymptotically exponential. The main step in the proof is a comparison lemma for the probability generating functions of Z(t) and of the embedded generation process.

We consider a diploid model of random selection. Assuming that the selective fitness is of the SAS–CFF type introduced by Gillespie, we obtain a diffusion approximation. We also discuss the existence of stationary distributions and recurrent properties.

Consider a two-type Moran population of size N with selection and mutation, where the selective advantage of the fit individuals is amplified at extreme environmental conditions. Assume selection and mutation are weak with respect to N, and extreme environmental conditions rarely occur. We show that, as $N\to\infty$, the type frequency process with time sped up by N converges to the solution to a Wright–Fisher-type SDE with a jump term modeling the effect of the environment. We use an extension of the ancestral selection graph (ASG) to describe the genealogical picture of the model. Next, we show that the type frequency process and the line-counting process of a pruned version of the ASG satisfy a moment duality. This relation yields a characterization of the asymptotic type distribution. We characterize the ancestral type distribution using an alternative pruning of the ASG. Most of our results are stated in annealed and quenched form.

In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.