Consider the process which starts with N ≥ 3 distinct points on ℝd, and fix a positive integer K < N. Of the total N points keep those N - K which minimize the energy amongst all the possible subsets of size N - K, and then replace the removed points by K independent and identically distributed points sampled according to some fixed distribution ζ. Repeat this process ad infinitum. We obtain various quite nonrestrictive conditions under which the set of points converges to a certain limit. This is a very substantial generalization of the `Keynesian beauty contest process' introduced in Grinfeld et al. (2015), where K = 1 and the distribution ζ was uniform on the unit cube.

Let V be an n-set, and let X be a random variable taking values in the power-set of V. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots $ , where the $X_i$ are independent random variables with distribution given by X. The covering time T is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^t X_i=V$ . The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focused almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.

In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.

The practically important classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the classic Markov embedding problem are obtained in the process. To achieve our results, we need to employ various algebraic and geometric tools, including commutativity, permutation invariance, and convexity. Of particular relevance in several demarcation results are Markov matrices that are idempotents.

A complete and sufficient statistic is found for various stationary Poisson processes of compact sets with known primary grain. In the particular case of a segment process, the uniformly best unbiased estimator for the length density is the number of segments hitting the sampling window divided by a certain constant and multiplied by the mean segment length.

Affine processes possess the property that expectations of exponential affine transformations are given by a set of Riccati differential equations, which is the main feature of this popular class of processes. In this paper we generalise these results for expectations of more general transformations. This is of interest in, e.g. doubly stochastic Markov models, in particular in life insurance. When using affine processes for modelling the transition rates and interest rate, the results presented allow for easy calculation of transition probabilities and expected present values.

Whittle enunciated an important reduction principle in dynamic programming when he showed that under certain conditions optimal strategies for Markov decision processes (MDPs) placed in parallel to one another take actions in a way which is consistent with the optimal strategies for the individual MDPs. However, the necessary and sufficient conditions given by Whittle are by no means always satisfied. We explore the status of this computationally attractive reduction principle when these conditions fail.

The method of Skorokhod embedding, that is to say of coupling to Brownian motion, is used to establish the asymptotic behaviour of one-dimensional scattering processes.

Analytic approximations are derived for the distribution of the first crossing time of a straight-line boundary by a d-dimensional Bessel process and its discrete time analogue. The main ingredient for the approximations is the conditional probability that the process crossed the boundary before time m, given its location beneath the boundary at time m. The boundary crossing probability is of interest as the significance level and power of a sequential test comparing d+1 treatments using an O'Brien-Fleming (1979) stopping boundary (see Betensky 1996). Also, it is shown by DeLong (1980) to be the limiting distribution of a nonparametric test statistic for multiple regression. The approximations are compared with exact values from the literature and with values from a Monte Carlo simulation.

For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.

A statistical method for estimating the factorisation of the spectral density of a stationary point process is presented and asymptotic properties of the resulting estimates are derived. The estimated functions are of interest in the analysis of a self-exciting process and more generally in the problem of linear prediction, and can be viewed as an alternative second-order analysis of the process. Some examples are discussed.

In this article we prove local convergence for a Boolean model of shells conditioned by the noncovering of the origin towards the thick hyperplane Poisson process in the Euclidean space. The existing results of Hall as well as the convergence theorems proved by Paroux or Molchanov concerned the zero-width process and the connected component of the unfilled region of the origin. Our results deal with the convergence in any given window of the space, with the earlier results of Paroux and Molchanov as a corollary.

The theory of geometric probabilities is concerned with randomly generated geometric objects. The aim is to compute probabilities of certain geometric events or distributions of random variables defined in a geometric way. Very often the computation even of simple expectations is too difficult, and one has to be satisfied with establishing estimates and, if possible, sharp inequalities. In geometric probabilities, convex sets play a prominent role, since often the convexity assumptions simplify the situation considerably. Extremal problems for geometric probabilities involving convex bodies can sometimes be attacked successfully by using suitable integral-geometric transformations and then applying classical inequalities from the geometry of convex bodies, or known methods for obtaining such inequalities. Examples of such results are the topic of this paper.

We study the long-term behaviour of a random walker embedded in a growing sequence of graphs. We define a (generally non-Markovian) real-valued stochastic process, called the knowledge process, that represents the ratio between the number of vertices already visited by the walker and the current size of the graph. We mainly focus on the case where the underlying graph sequence is the growing sequence of complete graphs.

In this paper we uniformly approximate the trajectories of the Cox–Ingersoll–Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view, the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact, simulation of the CIR dynamics at some deterministic time grid.

We study the growth of a population of bacteria in a dynamical hostile environment corresponding to the immune system of the colonized organism. The immune cells evolve as subcritical open clusters of oriented percolation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the remaining empty space. We prove that the population of bacteria grows linearly when it survives. From this perspective, we build general tools to study dependent oriented percolation models issued from renormalization processes.

Dimension reduction of multivariate data was developed by Y. Guan for point processes with Gaussian random fields as covariates. The generalization to fibre and surface processes is straightforward. In inverse regression methods, we suggest slicing based on geometrical marks. An investigation of the properties of this method is presented in simulation studies of random marked sets. In a refined model for dimension reduction, the second-order central subspace is analyzed in detail. A real data pattern is tested for independence of a covariate.

Let {Xn}n∈ℕ be a Markov chain on a measurable space with transition kernel P, and let The Markov kernel P is here considered as a linear bounded operator on the weighted-supremum space associated with V. Then the combination of quasicompactness arguments with precise analysis of eigenelements of P allows us to estimate the geometric rate of convergence ρV(P) of {Xn}n∈ℕ to its invariant probability measure in operator norm on A general procedure to compute ρV(P) for discrete Markov random walks with identically distributed bounded increments is specified.