We consider the critical Galton–Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity.

This result brings a new perspective on Vatutin’s dichotomy, claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals.

We are interested in a fragmentation process. We observe fragments frozen when their sizes are less than $\varepsilon$ ($\varepsilon>0$). It is known (Bertoin and Martínez, 2005) that the empirical measure of these fragments converges in law, under some renormalization. Hoffmann and Krell (2011) showed a bound for the rate of convergence. Here, we show a central limit theorem, under some assumptions. This gives us an exact rate of convergence.

We propose a discrete-time discrete-space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the proposed approximation with the Brownian bridge correction is $O(n^{-2})$ in the case of $C^2$ boundaries and a uniform time grid with n steps.

We study large-deviation probabilities of Telecom processes appearing as limits in a critical regime of the infinite-source Poisson model elaborated by I. Kaj and M. Taqqu. We examine three different regimes of large deviations (LD) depending on the deviation level. A Telecom process $(Y_t)_{t \ge 0}$ scales as $t^{1/\gamma}$ , where t denotes time and $\gamma\in(1,2)$ is the key parameter of Y. We must distinguish moderate LD ${\mathbb P}(Y_t\ge y_t)$ with $t^{1/\gamma} \ll y_t \ll t$ , intermediate LD with $ y_t \approx t$ , and ultralarge LD with $ y_t \gg t$ . The results we obtain essentially depend on another parameter of Y, namely the resource distribution. We solve completely the cases of moderate and intermediate LD (the latter being the most technical one), whereas the ultralarge deviation asymptotics is found for the case of regularly varying distribution tails. In all the cases considered, the large-deviation level is essentially reached by the minimal necessary number of ‘service processes’.

We present an efficient algorithm to generate a discrete uniform distribution on a set of p elements using a biased random source for p prime. The algorithm generalizes Von Neumann’s method and improves the computational efficiency of Dijkstra’s method. In addition, the algorithm is extended to generate a discrete uniform distribution on any finite set based on the prime factorization of integers. The average running time of the proposed algorithm is overall sublinear: $\operatorname{O}\!(n/\log n)$.

We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.

Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, Rácz, and Rashtchian [10] used combinatorial methods to show that $\exp({\mathrm{O}} (k \log_{k} n))$ samples suffice to reconstruct a complete k-ary tree with n nodes with high probability. We provide an alternative proof of this result, which allows us to generalize it to a broader class of tree topologies and deletion models. In our proofs we introduce the notion of a subtrace, which enables us to connect with and generalize recent mean-based complex analytic algorithms for string trace reconstruction.

The study of threshold functions has a long history in random graph theory. It is known that the thresholds for minimum degree k, k-connectivity, as well as k-robustness coincide for a binomial random graph. In this paper we consider an inhomogeneous random graph model, which is obtained by including each possible edge independently with an individual probability. Based on an intuitive concept of neighborhood density, we show two sufficient conditions guaranteeing k-connectivity and k-robustness, respectively, which are asymptotically equivalent. Our framework sheds some light on extending uniform threshold values in homogeneous random graphs to threshold landscapes in inhomogeneous random graphs.

We determine the distributions of some random variables related to a simple model of an epidemic with contact tracing and cluster isolation. This enables us to apply general limit theorems for super-critical Crump–Mode–Jagers branching processes. Notably, we compute explicitly the asymptotic proportion of isolated clusters with a given size amongst all isolated clusters, conditionally on survival of the epidemic. Somewhat surprisingly, the latter differs from the distribution of the size of a typical cluster at the time of its detection, and we explain the reasons behind this seeming paradox.

We present a Markov chain example where non-reversibility and an added edge jointly improve mixing time. When a random edge is added to a cycle of n vertices and a Markov chain with a drift is introduced, we get a mixing time of $O(n^{3/2})$ with probability bounded away from 0. If only one of the two modifications were performed, the mixing time would stay $\Omega(n^2)$ .

Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on independent and identically distributed vertex weights and the Euclidean distance $|x-y|$ . Depending on the various parameters involved, we get a rich phase diagram. We study graph distance and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.

In this paper we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMPs) whose dynamic is constrained by the presence of a boundary. On reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMPs.

We introduce a variant of Shepp’s classical urn problem in which the optimal stopper does not know whether sampling from the urn is done with or without replacement. By considering the problem’s continuous-time analog, we provide bounds on the value function and, in the case of a balanced urn (with an equal number of each ball type), an explicit solution is found. Surprisingly, the optimal strategy for the balanced urn is the same as in the classical urn problem. However, the expected value upon stopping is lower due to the additional uncertainty present.

We study a sceptical rumour model on the non-negative integer line. The model starts with two spreaders at sites 0, 1 and sceptical ignorants at all other natural numbers. Then each sceptic transmits the rumour, independently, to the individuals within a random distance on its right after s/he receives the rumour from at least two different sources. We say that the process survives if the size of the set of vertices which heard the rumour in this fashion is infinite. We calculate the probability of survival exactly, and obtain some bounds for the tail distribution of the final range of the rumour among sceptics. We also prove that the rumour dies out among non-sceptics and sceptics, under the same condition.

We study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U-statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U-statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

We propose a new Kalikow decomposition for continuous-time multivariate counting processes, on potentially infinite networks. We prove the existence of such a decomposition in various cases. This decomposition allows us to derive simulation algorithms that hold either for stationary processes with potentially infinite network but bounded intensities, or for processes with unbounded intensities in a finite network and with empty past before zero. The Kalikow decomposition is not unique, and we discuss the choice of the decomposition in terms of algorithmic efficiency in certain cases. We apply these methods to several examples: the linear Hawkes process, the age-dependent Hawkes process, the exponential Hawkes process, and the Galves–Löcherbach process.

The Bruss–Robertson–Steele (BRS) inequality bounds the expected number of items of random size which can be packed into a given suitcase. Remarkably, no independence assumptions are needed on the random sizes, which points to a simple explanation; the inequality is the integrated form of an $\omega$-by-$\omega$ inequality, as this note proves.

We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).

We establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.

Given a finite strongly connected directed graph $G=(V, E)$ , we study a Markov chain taking values on the space of probability measures on V. The chain, motivated by biological applications in the context of stochastic population dynamics, is characterized by transitions between states that respect the structure superimposed by E: mass (probability) can only be moved between neighbors in G. We provide conditions for the ergodicity of the chain. In a simple, symmetric case, we fully characterize the invariant probability.