We consider parallel single-server queues in heavy traffic with randomly split Hawkes arrival processes. The service times are assumed to be independent and identically distributed (i.i.d.) in each queue and are independent in different queues. In the critically loaded regime at each queue, it is shown that the diffusion-scaled queueing and workload processes converge to a multidimensional reflected Brownian motion in the non-negative orthant with orthonormal reflections. For the model with abandonment, we also show that the corresponding limit is a multidimensional reflected Ornstein–Uhlenbeck diffusion in the non-negative orthant.

We use Stein’s method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of samples arising from random events driven by a marked Poisson point process on $\mathbb{R}^d$. As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilisation and moment conditions. At the cost of an additional non-singularity condition, we show that the rates are in line with those under the Kolmogorov distance. We demonstrate the use of the theorems in four applications: Voronoi tessellations, k-nearest-neighbours graphs, timber volume, and maximal layers.

We consider infinitely wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a heavy-tailed distribution in the domain of attraction of a symmetric $\alpha$-stable distribution, where $\alpha\in(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $\alpha$-stable distribution having the same $\alpha$ parameter as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we extend a recent result of Favaro, Fortini, and Peluchetti (2020) to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $\alpha$-stable distributions, $\alpha\in(0,2]$.

We show how convergence to the Gumbel distribution in an extreme value setting can be understood in an information-theoretic sense. We introduce a new type of score function which behaves well under the maximum operation, and which implies simple expressions for entropy and relative entropy. We show that, assuming certain properties of the von Mises representation, convergence to the Gumbel distribution can be proved in the strong sense of relative entropy.

This paper investigates properties of the class of graphs based on exchangeable point processes. We provide asymptotic expressions for the number of edges, number of nodes, and degree distributions, identifying four regimes: (i) a dense regime, (ii) a sparse, almost dense regime, (iii) a sparse regime with power-law behaviour, and (iv) an almost extremely sparse regime. We show that, under mild assumptions, both the global and local clustering coefficients converge to constants which may or may not be the same. We also derive a central limit theorem for subgraph counts and for the number of nodes. Finally, we propose a class of models within this framework where one can separately control the latent structure and the global sparsity/power-law properties of the graph.

In this paper, we consider the convergence rate with respect to Wasserstein distance in the invariance principle for deterministic non-uniformly hyperbolic systems. Our results apply to uniformly hyperbolic systems and large classes of non-uniformly hyperbolic systems including intermittent maps, Viana maps, unimodal maps and others. Furthermore, as a non-trivial application to the homogenization problem, we investigate the Wasserstein convergence rate of a fast–slow discrete deterministic system to a stochastic differential equation.

A system of interacting multi-class finite-state jump processes is analyzed. The model under consideration consists of a block-structured network with dynamically changing multi-color nodes. The interactions are local and described through local empirical measures. Two levels of heterogeneity are considered: between and within the blocks where the nodes are labeled into two types. The central nodes are those connected only to nodes from the same block, whereas the peripheral nodes are connected to both nodes from the same block and nodes from other blocks. Limits of such systems as the number of nodes tends to infinity are investigated. In particular, under specific regularity conditions, propagation of chaos and the law of large numbers are established in a multi-population setting. Moreover, it is shown that, as the number of nodes goes to infinity, the behavior of the system can be represented by the solution of a McKean–Vlasov system. Then, we prove large deviations principles for the vectors of empirical measures and the empirical processes, which extends the classical results of Dawson and Gärtner (Stochastics 20, 1987) and Léonard (Ann. Inst. H. Poincaré Prob. Statist. 31, 1995).

We study homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.

We study the large-volume asymptotics of the sum of power-weighted edge lengths $\sum_{e \in E}|e|^\alpha$ in Poisson-based spatial random networks. In the regime $\alpha > d$, we provide a set of sufficient conditions under which the upper-large-deviation asymptotics are characterized by a condensation phenomenon, meaning that the excess is caused by a negligible portion of Poisson points. Moreover, the rate function can be expressed through a concrete optimization problem. This framework encompasses in particular directed, bidirected, and undirected variants of the k-nearest-neighbor graph, as well as suitable $\beta$-skeletons.

In this paper we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMPs) whose dynamic is constrained by the presence of a boundary. On reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMPs.

The basic idea of voting protocols is that nodes query a sample of other nodes and adjust their own opinion throughout several rounds based on the proportion of the sampled opinions. In the classic model, it is assumed that all nodes have the same weight. We study voting protocols for heterogeneous weights with respect to fairness. A voting protocol is fair if the influence on the eventual outcome of a given participant is linear in its weight. Previous work used sampling with replacement to construct a fair voting scheme. However, it was shown that using greedy sampling, i.e., sampling with replacement until a given number of distinct elements is chosen, turns out to be more robust and performant.

In this paper, we study fairness of voting protocols with greedy sampling and propose a voting scheme that is asymptotically fair for a broad class of weight distributions. We complement our theoretical findings with numerical results and present several open questions and conjectures.

We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure-preserving $\mathbb {Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a nullsequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third, and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on ${\mathbb Z}^{d}$-systems.

An edge flipping is a non-reversible Markov chain on a given connected graph, as defined in Chung and Graham (2012). In the same paper, edge flipping eigenvalues and stationary distributions for some classes of graphs were identified. We further study edge flipping spectral properties to show a lower bound for the rate of convergence in the case of regular graphs. Moreover, we show by a coupling argument that a cutoff occurs at $\frac{1}{4} n \log n$ for the edge flipping on the complete graph.

We study the coupon collector’s problem with group drawings. Assume there are n different coupons. At each time precisely s of the n coupons are drawn, where all choices are supposed to have equal probability. The focus lies on the fluctuations, as $n\to\infty$, of the number $Z_{n,s}(k_n)$ of coupons that have not been drawn in the first $k_n$ drawings. Using a size-biased coupling construction together with Stein’s method for normal approximation, a quantitative central limit theorem for $Z_{n,s}(k_n)$ is shown for the case that $k_n=({n/s})(\alpha\log(n)+x)$, where $0<\alpha<1$ and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $\alpha=1$, again using Stein’s method.

Large deviations of the largest and smallest eigenvalues of $\mathbf{X}\mathbf{X}^\top/n$ are studied in this note, where $\mathbf{X}_{p\times n}$ is a $p\times n$ random matrix with independent and identically distributed (i.i.d.) sub-Gaussian entries. The assumption imposed on the dimension size p and the sample size n is $p=p(n)\rightarrow\infty$ with $p(n)={\mathrm{o}}(n)$. This study generalizes one result obtained in [3].

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape, and we show that the shape is a Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. In the latter case we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

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.

Consider a set of n vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$, and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations and the vertex weights.

Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $\mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are independent and identically distributed copies of limiting vertex weights. Our set-up covers many sparse geometric random graph models from the literature, including geometric inhomogeneous random graphs (GIRGs), hyperbolic random graphs, continuum scale-free percolation, and weight-dependent random connection models.

We prove that the limiting degree distribution is mixed Poisson and the typical degree sequence is uniformly integrable, and we obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a byproduct of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

Two ensembles are frequently used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. Various examples have been analysed in the literature. It was found that BEE is the rule rather than the exception for two classes of constraints: sparse random graphs when the number of constraints is of the order of the number of vertices, and dense random graphs when there are two or more constraints that are frustrated. We establish BEE for a third class: dense random graphs with a single constraint on the density of a given simple graph. We show that BEE occurs in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We also show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as the spectral signature of BEE. We further show that in the replica symmetric region of the BEE-phase, BEE is due to the coexistence of two densities in the canonical ensemble.