The paper deals with large trunk line systems of the type appearing in telephone networks. There are many nodes or input sources, each pair of which is connected by a trunk line containing many individual circuits. Traffic arriving at either end of a trunk line wishes to communicate to the node at the other end. If the direct route is full, a rerouting might be attempted via an alternative route containing several trunks and connecting the same endpoints. The basic questions concern whether to reroute, and if so how to choose the alternative path. If the network is ‘large’ and fully connected, then the overflow traffic which is offered for rerouting to any trunk comes from many other trunks in the network with no one dominating. In this case one expects that some sort of averaging method can be used to approximate the rerouting requests and hence simplify the analysis. Essentially, the overflow traffic that a trunk offers the network for rerouting is in some average sense similar to the overflow traffic offered to that trunk. Indeed, a formalization of this idea involves the widely used (but generally heuristic) ‘fixed point' approximation method. One sets up the fixed point equations for appropriate rerouting strategies and then solves them to obtain an approximation to the system loss. In this paper we work in the heavy traffic regime, where the external offered traffic to any trunk is close to the service capacity of that trunk. It is shown that, as the number of links and circuits within each link go to infinity and for a variety of rerouting strategies, the system can be represented by an averaged limit. This limit is a reflected diffusion of the McKean–Vlasov (propagation of chaos) type, where the driving terms depend on the mean values of the solution of the equation. The averages occur due to the symmetry of the network and the averaging effects of the many interactions. This provides a partial justification for the fixed point method. The concrete dynamical systems flavor of the approach and the representations of the limit processes provide a useful way of visualizing the system and promise to be useful for the development of numerical methods and further analysis.

This meeting was envisaged by the organisers as the first of a continuing series of short, informal conferences within the general area of applied stochastic processes. The twenty-seven participants were drawn mostly from statistics and mathematics departments of British Universities, but also included a small number of biologists and geographers. The meeting was notable for its pleasantly informal atmosphere, and the consequent lively interchange of ideas, although for future meetings in the series a better balance between statisticians and applied scientists should, we feel, be effected.

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 consider a class of Gaussian processes which are obtained as height processes of some (d + 1)-dimensional dynamic random interface model on ℤd. We give an estimate of persistence probability, namely, large T asymptotics of the probability that the process does not exceed a fixed level up to time T. The interaction of the model affects the persistence probability and its asymptotics changes depending on the dimension d.

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.

Let ℬs([a,b];μ1,μ2,…,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,…,μs-1 as their first s-1 moments. In this paper we examine some aspects of the structure of ℬs([a,b];μ1,μ2,…,μs-1) and of the s-convex stochastic extrema in it. Using representation results of moment matrices à la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in ℬs([a,b];μ1,μ2,…,μs-1) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in ℬs([a,b];μ1,μ2,…,μs-1) by one of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.

Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.

We study a type of nonnormal small jump approximation of infinitely divisible distributions. By incorporating compound Poisson, gamma, and normal distributions, the approximation has a higher order of cumulant matching than its normal counterpart, and, hence, in many cases a higher rate of approximation error decay as the cutoff for the jump size tends to 0. The parameters of the approximation are easy to fix, and its random sampling has the same order of computational complexity as the normal approximation. An error bound of the approximation in terms of the total variation distance is derived. Simulations empirically show that the nonnormal approximation can have a significantly smaller error than its normal counterpart.

Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb{X}$ . Let $ S_n = \sum_{k=1}^n \ln f_{X_k}^{\prime}(1)$ be the Markov walk associated to $(X_n)_{n\geq 0}$ , where $f_i$ is the offspring generating function when the environment is $i \in \mathbb{X}$ . Conditioned on the event $\{ Z_n>0\}$ , we show the nondegeneracy of the limit law of the normalized number of particles ${Z_n}/{e^{S_n}}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}} $ jointly with $X_n$ . Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $ \log Z_n$ and $X_n$ given $Z_n>0$ .

Following the approach of standard filtering theory, we analyse investor valuation of firms, when these are modelled as geometric-Brownian state processes that are privately and partially observed, at random (Poisson) times, by agents. Tasked with disclosing forecast values, agents are able purposefully to withhold their observations; explicit filtering formulae are derived for downgrading the valuations in the absence of disclosures. The analysis is conducted for both a solitary firm and m co-dependent firms.

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 examine necessary and sufficient conditions for recurrence and positive recurrence of a class of irreducible, level-dependent quasi-birth-and-death (LDQBD) processes with a block tridiagonal structure that exhibits asymptotic convergence in the rows as the level tends to infinity. These conditions are obtained by exploiting a multi-dimensional Lyapunov drift approach, along with the theory of generalized Markov group inverses. Additionally, we highlight analogies to well-known average drift results for level-independent quasi-birth-and-death (QBD) processes.

Foschini gave a lower bound for the channel capacity of an N-transmit M-receive antenna system in a Raleigh fading environment with independence at both transmitters and receivers. We show that this bound is approximately normal.

A continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$ , a CBI $(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ . When $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ , the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ , immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).

We present an affine-invariant random walk for drawing uniform random samples from a convex body $\mathcal{K} \subset \mathbb{R}^n$ that uses maximum-volume inscribed ellipsoids, known as John’s ellipsoids, for the proposal distribution. Our algorithm makes steps using uniform sampling from the John’s ellipsoid of the symmetrization of $\mathcal{K}$ at the current point. We show that from a warm start, the random walk mixes in ${\widetilde{O}}\!\left(n^7\right)$ steps, where the log factors hidden in the ${\widetilde{O}}$ depend only on constants associated with the warm start and desired total variation distance to uniformity. We also prove polynomial mixing bounds starting from any fixed point x such that for any chord pq of $\mathcal{K}$ containing x, $\left|\log \frac{|p-x|}{|q-x|}\right|$ is bounded above by a polynomial in n.

We consider a nonnegative discrete time and bounded horizon process X for which 0 is an absorbing state and extend it by a random variable that is independent of X. We find a sufficient condition for the resulting process to satisfy, after a canonical time rescaling, the hypothesis of the monotone case theorem. If X describes a secretary type search on a poset with one maximal element or if we consider X with no extension then this condition assumes an especially simple log-concavity type form.

Generalized local mean normal measures μz, z ∈ Rd, are introduced for a nonstationary process X of convex particles. For processes with strictly convex particles it is then shown that X is weakly stationary and weakly isotropic if and only if μz is rotation invariant for all z ∈ Rd. The paper is concluded by extending this result to processes of cylinders, generalizing Theorem 1 of Schneider (2003).

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, $\alpha\in(0,2]$.