We study the integrated telegraph process $X_t$ under the assumption of general distribution for the random times between consecutive reversals of direction. Specifically, $X_t$ represents the position, at time t, of a particle moving U time units upwards with velocity c and D time units downwards with velocity $-c$. The latter motions are repeated cyclically, according to independent alternating renewals. Explicit expressions for the probability law of $X_t$ are given in the following cases: (i) (U, D) gamma-distributed; (ii) U exponentially distributed and D gamma-distributed. For certain values of the parameters involved, the probability law of $X_t$ is provided in a closed form. Some expressions for the moment generating function of $X_t$ and its Laplace transform are also obtained. The latter allows us to prove the existence of a Kac-type condition under which the probability density function of the integrated telegraph process, with identically distributed gamma intertimes, converges to that of the standard Brownian motion.

Finally, we consider the square of $X_t$ and disclose its distribution function, specifying the expression for some choices of the distribution of (U, D).

We consider residue expansions for survival and density/mass functions of first-passage distributions in finite-state semi-Markov processes (SMPs) in continuous and integer time. Conditions are given which guarantee that the residue expansions for these functions have a dominant exponential/geometric term. The key condition assumes that the relevant states for first passage contain an irreducible class, thus ensuring the same sort of dominant exponential/geometric terms as one gets for phase-type distributions in Markov processes. Essentially, the presence of an irreducible class along with some other conditions ensures that the boundary singularity b for the moment generating function (MGF) of the first-passage-time distribution is a simple pole. In the continuous-time setting we prove that b is a dominant pole, in that the MGF has no other pole on the vertical line $\{\text{Re}(s)=b\}.$ In integer time we prove that b is dominant if all holding-time mass functions for the SMP are aperiodic and non-degenerate. The expansions and pole characterisations address first passage to a single new state or a subset of new states, and first return to the starting state. Numerical examples demonstrate that the residue expansions are considerably more accurate than saddlepoint approximations and can provide a substitute for exact computation above the 75th percentile.

We consider two-dimensional Lévy processes reflected to stay in the positive quadrant. Our focus is on the non-standard regime when the mean of the free process is negative but the reflection vectors point away from the origin, so that the reflected process escapes to infinity along one of the axes. Under rather general conditions, it is shown that such behaviour is certain and each component can dominate the other with positive probability for any given starting position. Additionally, we establish the corresponding invariance principle providing justification for the use of the reflected Brownian motion as an approximate model. Focusing on the probability that the first component dominates, we derive a kernel equation for the respective Laplace transform in the starting position. This is done for the compound Poisson model with negative exponential jumps and, by means of approximation, for the Brownian model. Both equations are solved via boundary value problem analysis, which also yields the domination probability when starting at the origin. Finally, certain asymptotic analysis and numerical results are presented.

Let $X_t^\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$, $X_0=x$, killed at some terminal time T, where $Y_t$ is a Markov process having only jumps of length smaller than $\delta$, and $Z_t$ is a compound Poisson process with jumps of length bigger than $\delta$, for some fixed $\delta>0$. Under the assumptions that the summands in $Z_t$ are subexponential, we investigate the asymptotic behaviour of the potential function $u(x)= \mathbb{E}^x \int_0^\infty \ell\big(X_s^\sharp\big)ds$. The case of heavy-tailed entries in $Z_t$ corresponds to the case of ‘big claims’ in insurance models and is of practical interest. The main approach is based on the fact that u(x) satisfies a certain renewal equation.

The generalized perturbative approach is an all-purpose variant of Stein’s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications, leading, in each instance, to an extra log-factor vis-à-vis the rate in the independent case.

We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on n vertices with minimal degree linear in n is typically of order $\sqrt{n}$. A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.

We consider the unique measure of maximal entropy for proper 3-colorings of $\mathbb {Z}^{2}$, or equivalently, the so-called zero-slope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation-equivariant function of independent and identically distributed random variables placed on $\mathbb {Z}^{2}$. Along the way, we obtain various estimates on the mixing properties of this measure.

The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17] for the Kardar–Parisi–Zhang (KPZ) fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy–Widom distributions appear as special self-similar solutions of the KP and Korteweg–de Vries equations. In addition, it is noted that several known exact solutions of the KPZ equation also solve the KP equation.

We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature $\beta$. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.

Across a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call the ephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions tightly linked to the monotone cellular automata called bootstrap percolation. There are three classes of such models, the most studied being the critical one. In a recent series of works by Martinelli, Morris, Toninelli and the authors, it was shown that the KCM counterparts of critical bootstrap percolation models with the same properties split into two classes with different behaviour. Together with the companion paper by the first author, our work determines the logarithm of the infection time up to a constant factor for all critical KCM, which were previously known only up to logarithmic corrections. This improves all previous results except for the Duarte-KCM, for which we give a new proof of the best result known. We establish that on this level of precision critical KCM have to be classified into seven categories instead of the two in bootstrap percolation. In the present work, we establish lower bounds for critical KCM in a unified way, also recovering the universality result of Toninelli and the authors and the Duarte model result of Martinelli, Toninelli and the second author.

Signal-to-interference-plus-noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, independent and identically distributed, and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or more dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor $\gamma$ and the SINR threshold $\tau$ satisfy $\gamma \geq 1/(2\tau)$, then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.

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 an open discrete-time queueing network. We assume data is generated at nodes of the network as a discrete-time Bernoulli process. All nodes in the network maintain a queue and relay data, which is to be finally collected by a designated sink. We prove that the resulting multidimensional Markov chain representing the queue size of nodes has two behavior regimes depending on the value of the rate of data generation. In particular, we show that there is a nontrivial critical value of the data rate below which the chain is ergodic and converges to a stationary distribution and above which it is non-ergodic, i.e., the queues at the nodes grow in an unbounded manner. We show that the rate of convergence to stationarity is geometric in the subcritical regime.

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called ‘Poisson hypothesis’. However, in most applications this hypothesis is only conjectured. In this paper we establish the Poisson hypothesis for a general class of discrete-time, point-process-based dynamics that we propose to call fragmentation-interaction-aggregation processes, and which are introduced here. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete-time Galves–Löcherbach neural networks are used as a basic instance and illustration of our analysis.

In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

We investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

In a multitype branching process, it is assumed that immigrants arrive according to a non-homogeneous Poisson or a contagious Poisson process (both processes are formulated as a non-homogeneous birth process with an appropriate choice of transition intensities). We show that the normalized numbers of objects of the various types alive at time t for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, we provide some transient expectation results when there are only two types of particles.

We prove concentration inequality results for geometric graph properties of an instance of the Cooper–Frieze [5] preferential attachment model with edge-steps. More precisely, we investigate a random graph model that at each time $t\in \mathbb{N}$, with probability p adds a new vertex to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove concentration results for the global clustering coefficient as well as the clique number. More formally, we prove that the global clustering, with high probability, decays as $t^{-\gamma(p)}$ for a positive function $\gamma$ of p, whereas the clique number of these graphs is, up to subpolynomially small factors, of order $t^{(1-p)/(2-p)}$.

We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.