We introduce and study the class of branching-stable point measures, which can be seen as an analog of stable random variables when the branching mechanism for point measures replaces the usual addition. In contrast with the classical theory of stable (Lévy) processes, there exists a rich family of branching-stable point measures with a negative scaling exponent, which can be described as certain Crump‒Mode‒Jagers branching processes. We investigate the asymptotic behavior of their cumulative distribution functions, that is, the number of atoms in (-∞, x] as x→∞, and further depict the genealogical lineage of typical atoms. For both results, we rely crucially on the work of Biggins (1977), (1992).

We consider a special version of random sequential adsorption (RSA) with nearest-neighbor interaction on infinite tree graphs. In classical RSA, starting with a graph with initially inactive nodes, each of the nodes of the graph is inspected in a random order and is irreversibly activated if none of its nearest neighbors are active yet. We generalize this nearest-neighbor blocking effect to a degree-dependent threshold-based blocking effect. That is, each node of the graph is assumed to have its own degree-dependent threshold and if, upon inspection of a node, the number of active direct neighbors is less than that node's threshold, the node will become irreversibly active. We analyze the activation probability of nodes on an infinite tree graph, given the degree distribution of the tree and the degree-dependent thresholds. We also show how to calculate the correlation between the activity of nodes as a function of their distance. Finally, we propose an algorithm which can be used to solve the inverse problem of determining how to set the degree-dependent thresholds in infinite tree graphs in order to reach some desired activation probabilities.

We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

Consider a supercritical Crump‒Jagers process in which all births are at integer times (the lattice case). Let μ̂(z) be the generating function of the intensity of the offspring process, and consider the complex roots of μ̂(z)=1. The root of smallest absolute value is e-α=1∕m, where α>0 is the Malthusian parameter; let γ* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when γ*>e-α∕2=m-1∕2, they are asymptotically normal; (ii) when γ*=e-α∕2, they are still asymptotically normal, but with a larger variance; and (iii) when γ*<e-α∕2, the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Pólya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

Let P be the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vector πT, and let (n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n + 1) x (n + 1) northwest corner truncation of P arbitrarily, with invariant probability vector (n)πT. We derive computable V-norm bounds on the error between πT and (n)πT in terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 as n tends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.

We consider the evolution of the ancestral structure of a classical branching process in space and its diffusion limit. We also indicate how the conditional structure of the past can be described asymptotically in terms of suitable uniform Brownian trees.

We present a law of large numbers and a central limit theorem for the time to absorption of Λ-coalescents with dust started from n blocks, as n→∞. The proofs rely on an approximation of the logarithm of the block-counting process by means of a drifted subordinator.

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

Genealogical constructions of population processes provide models which simultaneously record the forward-in-time evolution of the population size (and distribution of locations and types for models that include them) and the backward-in-time genealogies of the individuals in the population at each time t. A genealogical construction for continuous-time Markov branching processes from Kurtz and Rodrigues (2011) is described and exploited to give the normalized limit in the supercritical case. A Seneta‒Heyde norming is identified as a solution of an ordinary differential equation. The analogous results are given for continuous-state branching processes, including proofs of the normalized limits of Grey (1974) in both the supercritical and critical/subcritical cases.

In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

We introduce a modified Galton‒Watson process using the framework of an infinite system of particles labelled by (x,t), where x is the rank of the particle born at time t. The key assumption concerning the offspring numbers of different particles is that they are independent, but their distributions may depend on the particle label (x,t). For the associated system of coupled monotone Markov chains, we address the issue of pathwise duality elucidated by a remarkable graphical representation in which the trajectories of the primary Markov chains and their duals coalesce to form forest graphs on a two-dimensional grid.