This paper studies estimation of stochastic block models with Rissanen’s minimum description length (MDL) principle in the dense graph asymptotics. We focus on the problem of model specification, i.e., identification of the number of blocks. Refinements of the true partition always decrease the code part corresponding to the edge placement, and thus a respective increase of the code part specifying the model should overweight that gain in order to yield a minimum at the true partition. The balance between these effects turns out to be delicate. We show that the MDL principle identifies the true partition among models whose relative block sizes are bounded away from zero. The results are extended to models with Poisson-distributed edge weights.

We study the evolution in time of the joint distribution of a pair of Feller processes, related by the fact that some random time ago they were identical, evolving as a single Feller process; from that time on, they began to evolve independently, conditional on a state at the time of split, according to the same Feller transition probabilities. Such processes are involved in the Fisher-Wright model: the distribution of the time counted backwards from the present to the time of split in the past is a function of deterministic but time-varying effective size 2N of the population from which the two processes are sampled. In terms of a corresponding family of Feller operators, assuming asymptotic stability or ergodicity of the process of mutation, we find the limit form of the distribution of such pairs of processes sampled from decaying, asymptotically constant, and growing populations. In the case where mutation is not asymptotically stable or ergodic, limit distributions are found for the distribution of relative differences.

We consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

In this paper, we introduce a slight variation of the dominated-coupling-from-the-past (DCFTP) algorithm of Kendall, for bounded Markov chains. It is based on the control of a (typically non-monotonic) stochastic recursion by another (typically monotonic) one. We show that this algorithm is particularly suitable for stochastic matching models with bounded patience, a class of models for which the steady-state distribution of the system is in general unknown in closed form. We first show that the Markov chain of this model can easily be controlled by an infinite-server queue. We then investigate the particular case where patience times are deterministic, and this control argument may fail. In that case we resort to an ad-hoc technique that can also be seen as a control (this time, by the arrival sequence). We then compare this algorithm to the primitive coupling-from-the-past (CFTP) algorithm and to control by an infinite-server queue, and show how our perfect simulation results can be used to estimate and compare, for instance, the loss probabilities of various systems in equilibrium.

We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝd by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.

We study the asymptotic behaviour of the expected exit time from an interval for the ARMA process, when the noise level approaches 0.

We adapt the classical definition of locally stationary processes in discrete time (see e.g. Dahlhaus, ‘Locally stationary processes’, in Time Series Analysis: Methods and Applications (2012)) to the continuous-time setting and obtain equivalent representations in the time and frequency domains. From this, a unique time-varying spectral density is derived using the Wigner–Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.