In this paper we focus on a class of linear Hawkes processes with general immigrants. These are counting processes with shot-noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of the age pyramid which evolves according to immigration and births. The virtue of this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.

We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi–Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees, with minor modification in the proof methods, done mutatis mutandis.

We study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω(x) when the inventory is at level x>0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω(x) is constant and ω(x) = a x. The key tool is the Wiener–Hopf factorization technique. When ω(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory.

Kuznetsov and co-authors in 2011‒14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further.

We study the convergence properties of a Monte Carlo estimator proposed in the physics literature to compute the quasi-stationary distribution on a transient set of a Markov chain (see De Oliveira and Dickman (2005), (2006), and Dickman and Vidigal (2002)). Using the theory of stochastic approximations we verify the consistency of the estimator and obtain an associated central limit theorem. We provide an example showing that convergence might occur very slowly if a certain eigenvalue condition is violated. We alleviate this problem using an easy-to-implement projection step combined with averaging.

We study G/G/∞ queues with renewal alternating service interruptions, where the service station experiences `up' and `down' periods. The system operates normally in the up periods, and all servers stop functioning while customers continue entering the system during the down periods. The amount of service a customer has received when an interruption occurs will be conserved and the service will resume when the down period ends. We use a two-parameter process to describe the system dynamics: Xr(t,y) tracking the number of customers in the system at time t that have residual service times strictly greater than y. The service times are assumed to satisfy either of the two conditions: they are independent and identically distributed with a distribution of a finite support, or are a stationary and weakly dependent sequence satisfying the ϕ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavy-traffic asymptotic regime where the arrival rate gets large and service time distribution is fixed, and the interruption down times are asymptotically negligible while the up times are of the same order as the service times. We show the functional law of large numbers and functional central limit theorem (FCLT) for the process Xr(t,y) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻, L1)) endowed with the Skorokhod M1 topology. The limit processes in the FCLT possess a stochastic decomposition property.

We consider Markov processes, which describe, e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady-state distribution of product form available in the literature, we note that this steady-state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one-step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for travelling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.

We perform an analysis of the potential time inhomogeneity in the dependence between multiple financial time series. To this end, we use the framework of copula theory and tackle the question of whether dependencies in such a case can be assumed constant throughout time or rather have to be modeled in a time-inhomogeneous way. We focus on parametric copula models and suitable inference techniques in the context of a special copula-based multivariate time series model. A recent result due to Chan et al. (2009) is used to derive the joint limiting distribution of local maximum-likelihood estimators on overlapping samples. By restricting the overlap to be fixed, we establish the limiting law of the maximum of the estimator series. Based on the limiting distributions, we develop statistical homogeneity tests, and investigate their local power properties. A Monte Carlo simulation study demonstrates that bootstrapped variance estimates are needed in finite samples. Empirical analyses on real-world financial data finally confirm that time-varying parameters are an exception rather than the rule.

In this paper we consider time-changed models of population evolution Xf(t) = X(Hf(t)), where X is a counting process and Hf is a subordinator with Laplace exponent f. In the case where X is a pure birth process, we study the form of the distribution, the intertimes between successive jumps, and the condition of explosion (also in the case of killed subordinators). We also investigate the case where X represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size n0. Finally, the subordinated linear birth–death process is considered. Special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.

In this paper we study a stochastic differential game between two insurers whose surplus processes are modelled by quadratic-linear diffusion processes. We consider an exit probability game. One insurer controls its risk process to minimize the probability that the surplus difference reaches a low level (indicating a disadvantaged surplus position of the insurer) before reaching a high level, while the other insurer aims to maximize the probability. We solve the game by finding the value function and the Nash equilibrium strategy in explicit forms.

In this paper we study a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward, and signed (positive and negative) switching costs. Using optimal stopping theory for discrete-parameter stochastic processes, we extend a well-known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs.

Starting with independent, identically distributed random variables X 1,X 2... and their partial sums (S n ), together with a nondecreasing sequence (b(n)), we consider the counting variable N=∑n 1(S n >b(n)) and aim for necessary and sufficient conditions on X 1 in order to obtain the existence of certain moments for N, as well as for generalized counting variables with weights, and multi-index random variables. The existence of the first moment of N when b(n)=εn, i.e. ∑n=1 ∞ℙ(|S n |>εn)<∞, corresponds to the notion of complete convergence as introduced by Hsu and Robbins in 1947 as a complement to Kolmogorov's strong law.

We describe in detail the speed of `coming down from infinity' for birth-and-death processes which eventually become extinct. Under general assumptions on the birth-and-death rates, we firstly determine the behavior of the successive hitting times of large integers. We identify two different regimes depending on whether the mean time for the process to go from n+1 to n is negligible or not compared to the mean time to reach n from ∞. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is gradual and a law of large numbers and a central limit theorem for the hitting times sequence hold. By an inversion procedure, we deduce that the process is almost surely equivalent to a nonincreasing function when the time goes to 0. Our results are illustrated by several examples including applications to population dynamics and population genetics. The particular case where the death rate varies regularly is studied in detail.

In this paper we consider linear functions constructed on two different weighted branching processes and provide explicit bounds for their Kantorovich–Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich–Rubinstein distance of linear functions. We focus on the case where the limits are endogenous fixed points of suitable smoothing transformations.

We provide exact computations for the drift of random walks in dependent random environments, including k-dependent and moving average environments. We show how the drift can be characterized and evaluated using Perron–Frobenius theory. Comparing random walks in various dependent environments, we demonstrate that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.

We consider an extension of the Poisson hail model where the service speed is either 0 or ∞ at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process, and on the geometrical properties of the zero speed set.

Branching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.

Let (X1,...,Xn) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and let Sn=eX1+⋯+eXn. The Laplace transform ℒ(θ)=𝔼e-θSn∝∫exp{-hθ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form and I(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacing hθ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙* is presented, and it is shown that I(θ)→ 1 as θ→∞. A variety of numerical methods for evaluating I(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density of Sn) are also given.

Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).

Let N and M be positive integers satisfying 1≤ M≤ N, and let 0< p0 < p1 < 1. Define a process {Xn}n=0∞ on ℤ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first N steps, the process behaves like a random walk that jumps to the right with probability p0 and to the left with probability 1-p0. At subsequent steps the jump mechanism is defined as follows: if at least M out of the N most recent jumps were to the right, then the probability of jumping to the right is p1; however, if fewer than M out of the N most recent jumps were to the right then the probability of jumping to the right is p0. We calculate the speed of the process. Then we let N→ ∞ and M/N→ r∈[0,1], and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number l of threshold levels, (Mi,pi) i=1l, above the pre-threshold level p0, as well as for one model with l=N such thresholds.