In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes.

In this paper we investigate gradient estimation for a class of contracting stochastic systems on a continuous state space. We find conditions on the one-step transitions, namely differentiability and contraction in a Wasserstein distance, that guarantee differentiability of stationary costs. Then we show how to estimate the derivatives, deriving an estimator that can be seen as a generalization of the forward sensitivity analysis method used in deterministic systems. We apply the results to examples, including a neural network model.

This article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$ , the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{{-}N\Lambda\}$ . The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

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.

This paper studies estimation of stochastic block models with Rissanen’s minimum description length (MDL) principle in the dense graph asymptotics. We focus on the problem of model specification, i.e., identification of the number of blocks. Refinements of the true partition always decrease the code part corresponding to the edge placement, and thus a respective increase of the code part specifying the model should overweight that gain in order to yield a minimum at the true partition. The balance between these effects turns out to be delicate. We show that the MDL principle identifies the true partition among models whose relative block sizes are bounded away from zero. The results are extended to models with Poisson-distributed edge weights.

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

Particles may be removed from a lattice by murder, coalescence, mutual annihilation and simple death. If the particle system is not to die out, the removed particles must be replaced by births. This letter shows that coalescence can be counteracted by arbitrarily small birth-rates and contrasts this with the situations for annihilation and pure death where there are critical phenomena. The problem is unresolved for murder.

We consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

The invariant Galton–Watson (IGW) tree measures are a one-parameter family of critical Galton–Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are attractors of arbitrary critical Galton–Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.

An M/G/∞ queue that is cleared whenever the level Q is exceeded plays the role of the outstanding orders process in an inventory system. The stationary version of the process is examined and an interesting property of the variance is illustrated.

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature; that is, if the surplus process just before the ith arrival is at level u, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$ . The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance, and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving Lévy process, including the generalised hyperbolic, normal-gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure Lévy process, and to a Brownian-driven SDE in the case of the Lévy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for Lévy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo, expectation-maximisation, and sequential Monte Carlo.

In this paper, we introduce a slight variation of the dominated-coupling-from-the-past (DCFTP) algorithm of Kendall, for bounded Markov chains. It is based on the control of a (typically non-monotonic) stochastic recursion by another (typically monotonic) one. We show that this algorithm is particularly suitable for stochastic matching models with bounded patience, a class of models for which the steady-state distribution of the system is in general unknown in closed form. We first show that the Markov chain of this model can easily be controlled by an infinite-server queue. We then investigate the particular case where patience times are deterministic, and this control argument may fail. In that case we resort to an ad-hoc technique that can also be seen as a control (this time, by the arrival sequence). We then compare this algorithm to the primitive coupling-from-the-past (CFTP) algorithm and to control by an infinite-server queue, and show how our perfect simulation results can be used to estimate and compare, for instance, the loss probabilities of various systems in equilibrium.

We introduce and study a new model: zero-automatic queues. Roughly, zero-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable zero-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of zero-automatic queues, we recover the simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers.

We study the asymptotic behaviour of the expected exit time from an interval for the ARMA process, when the noise level approaches 0.

Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEFs having either quadratic or cubic variance functions (VFs) and others as well. Particular attention is paid to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF, and the NEF generated by Kummer distributions of type 2. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEFs. Various lemmas and propositions are needed to verify this condition for such NEFs. It should be pointed out, however, that our results are mainly applied to a mixture of two populations.

An axiomatic derivation of the Wright–Fisher–Kimura (wfk) diffusion model for genetic drift is given using a variational principle. This is analogous to the characterization of the standard normal distribution in terms of an isoperimetric problem in the calculus of variations where the integrand is Shannon's information measure. We, on the other hand, give a Fisher-type information-theoretic interpretation of the variational principle largely motivated by the geometric approach to statistical likelihood theory due to S. V. Huzurbazzar, B. R. Rao and A. W. F. Edwards. In our process theory, the Ricci curvature tensor plays the role of information matrix. Ultimately, it is proved to be a normalized matrix of second partial derivates of the pseudodensity associated with the Christoffel velocity field. The proofs use classical projective differential geometry and depend on previous work in this series on the geometry of random drift.

A general discussion and survey is provided of the characterization of the normal distribution by the identical distribution of linear forms. The first result dates to 1923 when Pólya showed that if X and Y are i.i.d. random variables satisfying certain conditions, and if aX + bY is distributed as (a2 + b2)1/2X, then X has a normal distribution. This result has been generalized in several directions. In addition to a recasting of some of the results, an extension in the multivariate case is provided.