We present a new and straightforward algorithm that simulates exact sample paths for a generalized stress-release process. The computation of the exact law of the joint inter-arrival times is detailed and used to derive this algorithm. Furthermore, the martingale generator of the process is derived, and induces theoretical moments which generalize some results of [3] and are used to demonstrate the validity of our simulation algorithm.

Branching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanisms for point processes. Here we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results obtained during the past decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.

Reaction networks are commonly used within the mathematical biology and mathematical chemistry communities to model the dynamics of interacting species. These models differ from the typical graphs found in random graph theory since their vertices are constructed from elementary building blocks, i.e. the species. We consider these networks in an Erdös–Rényi framework and, under suitable assumptions, derive a threshold function for the network to have a deficiency of zero, which is a property of great interest in the reaction network community. Specifically, if the number of species is denoted by n and the edge probability by $p_n$ , then we prove that the probability of a random binary network being deficiency zero converges to 1 if $p_n\ll r(n)$ as $n \to \infty$ , and converges to 0 if $p_n \gg r(n)$ as $n \to \infty$ , where $r(n)=\frac{1}{n^3}$ .

We study the so-called frog model on ${\mathbb{Z}}$ with two types of lazy frogs, with parameters $p_1,p_2\in (0,1]$ respectively, and a finite expected number of dormant frogs per site. We show that for any such $p_1$ and $p_2$ there is positive probability that the two types coexist (i.e. that both types activate infinitely many frogs). This answers a question of Deijfen, Hirscher, and Lopes in dimension one.

We investigate properties of random mappings whose core is composed of derangements as opposed to permutations. Such mappings arise as the natural framework for studying the Screaming Toes game described, for example, by Peter Cameron. This mapping differs from the classical case primarily in the behaviour of the small components, and a number of explicit results are provided to illustrate these differences.

Asymptotics deviation probabilities of the sum $S_n=X_1+\dots+X_n$ of independent and identically distributed real-valued random variables have been extensively investigated, in particular when $X_1$ is not exponentially integrable. For instance, Nagaev (1969a, 1969b) formulated exact asymptotics results for $\mathbb{P}(S_n>x_n)$ with $x_n\to \infty$ when $X_1$ has a semiexponential distribution. In the same setting, Brosset et al. (2020) derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large-deviation theory and making the rate function at the transition explicit. In this paper we exhibit the same asymptotic behavior for triangular arrays of semiexponentially distributed random variables.

We derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in $\mathbb{R}^d$ of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

We consider a variant of a classical coverage process, the Boolean model in $\mathbb{R}^d$ . Previous efforts have focused on convergence of the unoccupied region containing the origin to a well-studied limit C. We study the intersection of sets centered at points of a Poisson point process confined to the unit ball. Using a coupling between the intersection model and the original Boolean model, we show that the scaled intersection converges weakly to the same limit C. Along the way, we present some tools for studying statistics of a class of intersection models.

We prove a surprising symmetry between the law of the size $G_n$ of the greedy independent set on a uniform Cayley tree $ \mathcal{T}_n$ of size n and that of its complement. We show that $G_n$ has the same law as the number of vertices at even height in $ \mathcal{T}_n$ rooted at a uniform vertex. This enables us to compute the exact law of $G_n$ . We also give a Markovian construction of the greedy independent set, which highlights the symmetry of $G_n$ and whose proof uses a new Markovian exploration of rooted Cayley trees that is of independent interest.

An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of jth-generation individuals with birth times $\leq t$ , when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such generations form a subset of the set of intermediate generations.

We study supercritical branching processes under the influence of an independent and identically distributed (i.i.d.) emigration component. We provide conditions under which the lifetime of the process is finite or has a finite expectation. A theorem of Kesten–Stigum type is obtained, and the extinction probability for a large initial population size is related to the tail behaviour of the emigration.

In this paper we employ a Gaussian-type heat kernel estimate to establish Krylov’s estimate and Khasminskii’s estimate for the Euler–Maruyama (EM) algorithm. For applications, by taking Zvonkin’s transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz.

We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation of long time averages. We establish the consistency of weighted ensemble in this setting by proving an ergodic theorem for time averages. As part of the proof, we derive explicit variance formulas that could be useful for optimizing the method.

We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics, such as the clustering coefficient and assortativity, is determined by the local weak limit.

As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton–Watson branching process.

It is well known that each statistic in the family of power divergence statistics, across n trials and r classifications with index parameter $\lambda\in\mathbb{R}$ (the Pearson, likelihood ratio, and Freeman–Tukey statistics correspond to $\lambda=1,0,-1/2$ , respectively), is asymptotically chi-square distributed as the sample size tends to infinity. We obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample n, and all index parameters ( $\lambda>-1$ ) for which such finite-sample bounds are meaningful. We obtain bounds that are of the optimal order $n^{-1}$ . The dependence of our bounds on the index parameter $\lambda$ and the cell classification probabilities is also optimal, and the dependence on the number of cells is also respectable. Our bounds generalise, complement, and improve on recent results from the literature.

Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$ , let $Z_n(A)$ be the number of particles located in interval A at generation n. It is well known that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges almost surely to $\nu(A)$ as $n\rightarrow\infty$ , where $\nu$ is the standard Gaussian measure. We investigate its large-deviation probabilities under the condition that the step size or offspring law has a heavy tail, i.e. a decay rate of $\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$ as $n\rightarrow\infty$ , where $p\in(\nu(A),1)$ . Our results complete those in Chen and He (2019) and Louidor and Perkins (2015).

This paper is devoted to the study of regime-switching jump diffusion processes with countable regimes. It aims to establish Foster–Lyapunov-type criteria for exponential ergodicity of such processes. After recalling results concerning the petiteness of compact sets, this paper presents sufficient conditions for the existence of a Foster–Lyapunov function; this, in turn, helps to establish sufficient conditions for the desired exponential ergodicity for regime-switching jump diffusion processes. Finally, an application to feedback control problems is presented.

Simulated tempering is a popular method of allowing Markov chain Monte Carlo algorithms to move between modes of a multimodal target density $\pi$ . Tawn, Moores and Roberts (2021) introduces the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that, under appropriate assumptions, the ALPS algorithm mixes in time $O(d [\log d]^2)$ or O(d), depending on which version is used.

We analyse features of the patterns formed from a simple model for a martensitic phase transition that fragments the unit square into rectangles. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the lengths of the interfaces in the pattern, and we establish limit theorems for some of the asymptotics of the interface profile. In particular, we are able to use a general branching process to show almost sure power law decay of the number of interfaces of at least a certain size and a general branching random walk to examine the numbers of rectangles of a certain aspect ratio. In doing so we extend a theorem on the growth of the general branching random walk as well as developing results on the tail behaviour of the limiting random variable in our general branching process.