Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.

This work concerns Markov decision chains on a finite state space. The decision-maker has a constant and nonnull risk sensitivity coefficient, and the performance of a control policy is measured by two different indices, namely, the discounted and average criteria. Motivated by well-known results for the risk-neutral case, the problem of approximating the optimal risk-sensitive average cost in terms of the optimal risk-sensitive discounted value functions is addressed. Under suitable communication assumptions, it is shown that, as the discount factor increases to 1, appropriate normalizations of the optimal discounted value functions converge to the optimal average cost, and to the functional part of the solution of the risk-sensitive average cost optimality equation.

We discuss percolation and random walks in a class of homogeneous ultrametric spaces together with similarities and differences in ultrametric and Euclidean spaces. We briefly outline the role of these models in the study of interacting systems. Several open problems are presented.

The forest of mutations associated to a multitype branching forest is obtained by merging together all vertices in each of its clusters and by preserving connections between them. (Here, by cluster, we mean a maximal connected component of the forest in which all vertices have the same type.) We first show that the forest of mutations of any multitype branching forest is itself a branching forest. Then we give its progeny distribution and we describe some of its crucial properties in terms of the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to ∞ and when the number of roots tends to ∞. The continuous-time case is then investigated by considering multitype branching forests with edge lengths. When mutations are nonreversible, we give a representation of their emergence times which allows us to describe the asymptotic behaviour of the latter, under certain conditions on the mutation rates. These results have potential relevance for emergence of mutations in population cells, particularly for genetic evolution of cancer or development of infectious diseases.

One model of real-life spreading processes is that of first-passage percolation (also called the SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with independent and identically distributed heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow due to bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power-law distribution ℙ(ξ>t)∼t-α with infinite mean. For any finite connected graph G with a root s, we find the largest number of vertices κ(G,s) that are infected in finite expected time, and prove that for every k≤κ(G,s), the expected time to infect k vertices is at most O(k1/α). Then we show that adding a single edge from s to a random vertex in a random tree 𝒯 typically increases κ(𝒯,s) from a bounded variable to a fraction of the size of 𝒯, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton--Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős‒Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

Recent progress in microdissection and in DNA sequencing has facilitated the subsampling of multi-focal cancers in organs such as the liver in several hundred spots, helping to determine the pattern of mutations in each of these spots. This has led to the construction of genealogies of the primary, secondary, tertiary, and so forth, foci of the tumor. These studies have led to diverse conclusions concerning the Darwinian (selective) or neutral evolution in cancer. Mathematical models of the development of multi-focal tumors have been devised to support these claims. We offer a model for the development of a multi-focal tumor: it is a mathematically rigorous refinement of a model of Ling et al. (2015). Guided by numerical studies and simulations, we show that the rigorous model, in the form of an infinite-type branching process, displays distributions of tumor size which have heavy tails and moments that become infinite in finite time. To demonstrate these points, we obtain bounds on the tails of the distributions of the process and an infinite series expansion for the first moments. In addition to its inherent mathematical interest, the model is corroborated by recent literature on apparent super-exponential growth in cancer metastases.

We investigate the asymptotic properties of the transient queue length process Q(t)=max(Q(0)+X(t)-ct, sup0≤s≤t(X(t)-X(s)-c(t-s))), t≥0, in the Gaussian fluid queueing model, where the input process X is modeled by a centered Gaussian process with stationary increments and c>0 is the output rate. More specifically, under some mild conditions on X and Q(0)=x≥0, we derive the exact asymptotics of πx,Tu(u)=ℙ(Q(Tu)>u) as u→∞. The interplay between u and Tu leads to two qualitatively different regimes: short-time horizon when Tu is relatively small with respect to u, and moderate- or long-time horizon when Tu is asymptotically much larger than u. As a by-product, we discuss the implications for the speed of convergence to stationarity of the model studied.

The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z > 0 and law of radii Q. The formal unnormalised density is given by q N cc , where q is a fixed parameter and N cc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an α-stable law with 1 < α < 2. We prove that the derivative martingale Dn converges to a nontrivial limit D∞ under some regular conditions. We also study the additive martingale Wn and prove that n1/αWn converges in probability to a constant multiple of D∞.

We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.

The Holme‒Kim random graph process is a variant of the Barabási‒Álbert scale-free graph that was designed to exhibit clustering. In this paper we show that whether the model does indeed exhibit clustering depends on how we define the clustering coefficient. In fact, we find that the local clustering coefficient typically remains positive whereas global clustering tends to 0 at a slow rate. These and other results are proven via martingale techniques, such as Freedman's concentration inequality combined with a bootstrapping argument.

We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.

We consider a Markov-modulated fluid flow production model under the D-policy, that is, as soon as the storage reaches level 0, the machine becomes idle until the total storage exceeds a predetermined threshold D. Thus, the production process alternates between a busy and an idle machine. During the busy period, the storage decreases linearly due to continuous production and increases due to supply; during the idle period, no production is rendered by the machine and the storage level increases by only supply arrivals. We consider two types of model with different supply process patterns: continuous inflows with linear rates (fluid type), and batch inflows, where the supplies arrive according to a Markov additive process (MAP) and their sizes are independent and have phase-type distributions depending on the type of arrival (MAP type). Four types of cost are considered: a setup cost, a production cost, a penalty cost for an idle machine, and a storage cost. Using tools from multidimensional martingale and hitting time theory, we derive explicit formulae for these cost functionals in the discounted case. Numerical examples, a sensitivity analysis, and insights are provided.

The largest components of the critical Erdős–Rényi graph, G(n, p) with p = 1 / n, have size of order n2/3 with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size an2/3 for large a. Our results, which extend the work of Pittel (2001), allow a to depend upon n and also hold for a range of values of p around 1 / n. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.

Let Sn,n≥1, be the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that Sn∕(nlog2n)→ℙ1 as n→∞. In this paper we review some earlier results of ours and extend some of them as we consider an asymmetric St. Petersburg game, in which the distribution of the payoff X is given by ℙ(X=srk-1)=pqk-1,k=1,2,…, where p+q=1 and s,r>0. Two main results are extensions of the Feller weak law and the convergence in distribution theorem of Martin-Löf (1985). Moreover, it is well known that almost-sure convergence fails, though Csörgő and Simons (1996) showed that almost-sure convergence holds for trimmed sums and also for sums trimmed by an arbitrary fixed number of maxima. In view of the discreteness of the distribution we focus on `max-trimmed sums', that is, on the sums trimmed by the random number of observations that are equal to the largest one, and prove limit theorems for simply trimmed sums, for max-trimmed sums, as well as for the `total maximum'. Analogues with respect to the random number of summands equal to the minimum are also obtained and, finally, for joint trimming.

The Quicksort process R (Rösler (2018)) can be characterized as the unique endogenous solution of the inhomogeneous stochastic fixed point equation R=D(UR1(1∧t∕U)+𝟭{U<t}(1-U)R2((t-U)∕(1-U))+C(U,t))t on the space 𝒟 of càdlàg functions, such that R(1) has the Quicksort distribution. In this paper we characterize all 𝒟-valued solutions of that equation. Every solution can be represented as the convolution of a solution of the inhomogeneous equation and a general solution of the homogeneous equation (Rüschendorf (2006)). The general solutions of the homogeneous equation are the distributions of Cauchy processes Y with constant drift. Any distribution of R+Y for independent R and Y is a solution of the inhomogeneous equation. Every solution of the inhomogeneous equation is of the form R+Y, where R and Y are independent. The endogenous solutions for the inhomogeneous equation are the shifted Quicksort process distributions. In comparison, the Quicksort distribution is the endogenous solution of the Quicksort fixed point equation unique up to a constant (Rösler (1991)). The general solution can be represented as the convolution of the shifted Quicksort distribution and some symmetric Cauchy distribution (Fill and Janson (2000)), possibly degenerate.

In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.

We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the important statistical content of such models is given by the normalization constant of the models, and, in particular, an appropriately scaled limit of the normalization, which is called the free energy. We derive precise asymptotics for the normalization constant for finite graphs. We use this to derive a formula for the free energy. The limit is analytic everywhere except along a curve corresponding to a first-order phase transition. We examine unusual behavior of the model along the phase transition curve.

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

We develop a forward-reverse expectation-maximization (FREM) algorithm for estimating parameters of a discrete-time Markov chain evolving through a certain measurable state-space. For the construction of the FREM method, we develop forward-reverse representations for Markov chains conditioned on a certain terminal state. We prove almost sure convergence of our algorithm for a Markov chain model with curved exponential family structure. On the numerical side, we carry out a complexity analysis of the forward-reverse algorithm by deriving its expected cost. Two application examples are discussed.