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.

Let $\theta $ be an irrational real number. The map $T_\theta : y \mapsto (y+\theta ) \,\mod \!\!\: 1$ from the unit interval $\mathbf {I} = [0,1[$ (endowed with the Lebesgue measure) to itself is ergodic. In 2002, Rudolph and Hoffman showed in [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2) 156(1) (2002), 79–101] that the measure-preserving map $[T_\theta ,\mathrm {Id}]$ is isomorphic to a one-sided dyadic Bernoulli shift. Their proof is not constructive. A few years before, Parry [Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] had provided an explicit isomorphism under the assumption that $\theta $ is extremely well approached by the rational numbers, namely,

$$ \begin{align*}\inf_{q \ge 1} q^44^{q^2}~\mathrm{dist}(\theta,q^{-1}\mathbb{Z}) = 0.\end{align*} $$

Whether the explicit map considered by Parry is an isomorphism or not in the general case was still an open question. In Leuridan [Bernoulliness of $[T,\mathrm {Id}]$ when T is an irrational rotation: towards an explicit isomorphism. Ergod. Th. & Dynam. Sys. 41(7) (2021), 2110–2135] we relaxed Parry’s condition into

$$ \begin{align*}\inf_{q \ge 1} q^4~\mathrm{dist}(\theta,q^{-1}\mathbb{Z}) = 0.\end{align*} $$

In the present paper, we remove the condition by showing that the explicit map considered by Parry is always an isomorphism. With a few adaptations, the same method works with $[T,T^{-1}]$.

We consider a stochastic SIR (susceptible $\rightarrow$ infective $\rightarrow$ removed) model in which the infectious periods are modulated by a collection of independent and identically distributed Feller processes. Each infected individual is associated with one of these processes, the trajectories of which determine the duration of his infectious period, his contamination rate, and his type of removal (e.g. death or immunization). We use a martingale approach to derive the distribution of the final epidemic size and severity for this model and provide some general examples. Next, we focus on a single infected individual facing a given number of susceptibles, and we determine the distribution of his outcome (number of contaminations, severity, type of removal). Using a discrete-time formulation of the model, we show that this distribution also provides us with an alternative, more stable method to compute the final epidemic outcome distribution.

We consider a one-dimensional superprocess with a supercritical local branching mechanism $\psi$, where particles move as a Brownian motion with drift $-\rho$ and are killed when they reach the origin. It is known that the process survives with positive probability if and only if $\rho<\sqrt{2\alpha}$, where $\alpha=-\psi'(0)$. When $\rho<\sqrt{2 \alpha}$, Kyprianou et al. [18] proved that $\lim_{t\to \infty}R_t/t =\sqrt{2\alpha}-\rho$ almost surely on the survival set, where $R_t$ is the rightmost position of the support at time t. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of $\mathbb{P}_{\delta_x} (R_t >\gamma t+\theta)$ as $t \to \infty$, where $\gamma >\sqrt{2 \alpha} -\rho$, $\theta \ge 0$. As a by-product, a related Yaglom-type conditional limit theorem is obtained. Analogous results for branching Brownian motion can be found in Harris et al. [13].

Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.

We construct a class of non-reversible Metropolis kernels as a multivariate extension of the guided-walk kernel proposed by Gustafson (Statist. Comput. 8, 1998). The main idea of our method is to introduce a projection that maps a state space to a totally ordered group. By using Haar measure, we construct a novel Markov kernel termed the Haar mixture kernel, which is of interest in its own right. This is achieved by inducing a topological structure to the totally ordered group. Our proposed method, the $\Delta$-guided Metropolis–Haar kernel, is constructed by using the Haar mixture kernel as a proposal kernel. The proposed non-reversible kernel is at least 10 times better than the random-walk Metropolis kernel and Hamiltonian Monte Carlo kernel for the logistic regression and a discretely observed stochastic process in terms of effective sample size per second.

We consider a spatial model of cancer in which cells are points on the d-dimensional torus $\mathcal{T}=[0,L]^d$, and each cell with $k-1$ mutations acquires a kth mutation at rate $\mu_k$. We assume that the mutation rates $\mu_k$ are increasing, and we find the asymptotic waiting time for the first cell to acquire k mutations as the torus volume tends to infinity. This paper generalizes results on waiting for $k\geq 3$ mutations in Foo et al. (2020), which considered the case in which all of the mutation rates $\mu_k$ are the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates.

In this paper, we analyze a two-queue random time-limited Markov-modulated polling model. In the first part of the paper, we investigate the fluid version: fluid arrives at the two queues as two independent flows with deterministic rate. There is a single server that serves both queues at constant speeds. The server spends an exponentially distributed amount of time in each queue. After the completion of such a visit time to one queue, the server instantly switches to the other queue, i.e., there is no switch-over time.

For this model, we first derive the Laplace–Stieltjes transform (LST) of the stationary marginal fluid content/workload at each queue. Subsequently, we derive a functional equation for the LST of the two-dimensional workload distribution that leads to a Riemann–Hilbert boundary value problem (BVP). After taking a heavy-traffic limit, and restricting ourselves to the symmetric case, the BVP simplifies and can be solved explicitly.

In the second part of the paper, allowing for more general (Lévy) input processes and server switching policies, we investigate the transient process limit of the joint workload in heavy traffic. Again solving a BVP, we determine the stationary distribution of the limiting process. We show that, in the symmetric case, this distribution coincides with our earlier solution of the BVP, implying that in this case the two limits (stationarity and heavy traffic) commute.

During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t \geq 0$, given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.

We consider a class of processes describing a population consisting of k types of individuals. The process is almost surely absorbed at the origin within finite time, and we study the expected time taken for such extinction to occur. We derive simple and precise asymptotic estimates for this expected persistence time, starting either from a single individual or from a quasi-equilibrium state, in the limit as a system size parameter N tends to infinity. Our process need not be a Markov process on $ {\mathbb Z}_+^k$; we allow the possibility that individuals’ lifetimes may follow more general distributions than the exponential distribution.

Let $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$. Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$, $n\geqslant 1$. For any $t \in {\mathbb R}$, denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$. By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$. We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

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 prove some estimates for the variations of transition probabilities of the (1+1)-affine process. From these estimates we deduce the strong Feller and the ergodic properties of the total variation distance of the process. The key strategy is the pathwise construction and analysis of several Markov couplings using strong solutions of stochastic equations.

Consider a two-type Moran population of size N with selection and mutation, where the selective advantage of the fit individuals is amplified at extreme environmental conditions. Assume selection and mutation are weak with respect to N, and extreme environmental conditions rarely occur. We show that, as $N\to\infty$, the type frequency process with time sped up by N converges to the solution to a Wright–Fisher-type SDE with a jump term modeling the effect of the environment. We use an extension of the ancestral selection graph (ASG) to describe the genealogical picture of the model. Next, we show that the type frequency process and the line-counting process of a pruned version of the ASG satisfy a moment duality. This relation yields a characterization of the asymptotic type distribution. We characterize the ancestral type distribution using an alternative pruning of the ASG. Most of our results are stated in annealed and quenched form.

This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose time-inhomogeneity is asymptotically periodic. Under a Lyapunov/minorization condition, it is shown that, for any measurable bounded function f, the time average $\frac{1}{t} \int_0^t f(X_s)ds$ converges in $\mathbb{L}^2$ towards a limiting distribution, starting from any initial distribution for the process $(X_t)_{t \geq 0}$ . This convergence can be improved to an almost sure convergence under an additional assumption on the initial measure. This result is then applied to show the existence of a quasi-ergodic distribution for processes absorbed by an asymptotically periodic moving boundary, satisfying a conditional Doeblin condition.

We study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function, which is a function of its weight and degree, and connects to $\ell$ new-coming vertices. Under a certain technical assumption, applying the theory of Crump–Mode–Jagers branching processes, we derive formulas for the limiting distributions of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we rigorously prove observations of Bianconi related to the evolving Cayley tree (Phys. Rev. E 66, paper no. 036116, 2002). We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call ‘generalised preferential attachment with fitness’. We show that this model can exhibit condensation, where a positive proportion of edges accumulates around vertices with maximal weight, or, more drastically, can have a degenerate limiting degree distribution, where the entire proportion of edges accumulates around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process.

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 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.

Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$-valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$, $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$, where $\{\vartheta_n\}_{n \ge 1}$ are independent and identically distributed random variables with common probability distribution $\mathfrak{p}$, $F(\cdot,\cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of $\{\vartheta_n\}_{n \ge 1}$. Under parametric perturbation of both F and $\mathfrak{p}$, we are interested in the robustness of the V-geometrical ergodicity property of $\{X_n\}_{n\in{\mathbb{N}}}$, of its invariant probability measure, and finally of the probability distribution of $X_n$. Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $\Omega ((n/k) \log n)$ on the stationary cover time, holding for any $n$ -vertex graph $G$ and any $1 \leq k =o(n\log n )$ . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.