The concept of isotropically random mosaic introduced by Pielou, is recalled. The notion of anisotropically random mosaic with respect to a point is defined. An example is given.

Let be i.i.d. uniform on (0,1) random variables and define Si,n = Ui,n–1Ui–1,n–1, i = 1, · ··, n where the Ui–n–1 are the order statistics from a sample of size n – 1 and U0,n–1 =0 and Un,n–1 = 1. The Si,n are called the spacings divided by U1,· ··,Un–1. For a fixed integer l, set . Exact and weak limit results are obtained for the Ml,n. Further we show that with probability 1 This extends results of Cheng.

A sequence of independent, identically distributed random variables is observed. After a sample of m has been collected attention is fixed to a particular observation. In this paper the fluctuations of the rank of this observation, as sampling continues, will be studied.

The process can be modelled by a Pólya urn scheme, and new results are obtained which are of interest in both sequential rank and Pólya urn contexts.

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold:

1. (a) no server sits idle while there are customers waiting to be served;

2. (b) the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one);

The paper studies a network of queues in which units arrive singly in a Poisson stream at a service channel S from which they branch out into k parallel channels S1, S2, …, Sk. After having been serviced at S1, S2, … Sk, the units converge again into a single channel S′. The service times of units at each of the channels are assumed to be exponential. Units finally serviced at S’ may leave the system or may again join S. This has been considered by taking two models denoted as Model A and Model B. Steady-state probabilities giving the number of units present in the system have been obtained explicitly for both the models. The expressions for mean queue lengths have also been arrived at.

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.

In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.

Let X1, X2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).

Consider a random walk Sn = ∑i=0nXi with negative drift. This paper deals with upper bounds for the maximum M = maxn≥1Sn of this random walk in different settings of power moment existences. As is usual for deriving upper bounds, we truncate summands. Therefore, we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby, we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for Mτ = maxn≤τSn. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

This paper analyzes the training process of generative adversarial networks (GANs) via stochastic differential equations (SDEs). It first establishes SDE approximations for the training of GANs under stochastic gradient algorithms, with precise error bound analysis. It then describes the long-run behavior of GAN training via the invariant measures of its SDE approximations under proper conditions. This work builds a theoretical foundation for GAN training and provides analytical tools to study its evolution and stability.

In this paper we consider the problem of simultaneously estimating rare-event probabilities for a class of Gaussian random fields. A conventional rare-event simulation method is usually tailored to a specific rare event and consequently would lose estimation efficiency for different events of interest, which often results in additional computational cost in such simultaneous estimation problems. To overcome this issue, we propose a uniformly efficient estimator for a general family of Hölder continuous Gaussian random fields. We establish the asymptotic and uniform efficiency of the proposed method and also conduct simulation studies to illustrate its effectiveness.

The equilibrium delay distribution for a system with finite-source input, constant service time, and s > 1 servers, as determined by simulation, showed the existence of values of t such that the probability of delay greater than t is a non-monotone function of the input intensity. Since repeated efforts to remove this counter-intuitive result by debugging failed, a simple case was attacked by analytical and numerical methods; and the same anomaly was revealed. The method of analysis is described and an explanation of the result is offered.

Exact simulation approaches for a class of diffusion bridges have recently been proposed based on rejection sampling techniques. The existing rejection sampling methods may not be practical owing to small acceptance probabilities. In this paper we propose an adaptive approach that improves the existing methods significantly under certain scenarios. The idea of the new method is based on a layered process, which can be simulated from a layered Brownian motion with reweighted layer probabilities. We will show that the new exact simulation method is more efficient than existing methods theoretically and via simulation.

Real networks often exhibit clustering, the tendency to form relatively small groups of nodes with high edge densities. This clustering property can cause large numbers of small and dense subgraphs to emerge in otherwise sparse networks. Subgraph counts are an important and commonly used source of information about the network structure and function. We study probability distributions of subgraph counts in a community affiliation graph. This is a random graph generated as an overlay of m partly overlapping independent Bernoulli random graphs (layers) $G_1,\dots,G_m$ with variable sizes and densities. The model is parameterised by a joint distribution of layer sizes and densities. When m grows linearly in the number of nodes n, the model generates sparse random graphs with a rich statistical structure, admitting a nonvanishing clustering coefficient and a power-law limiting degree distribution. In this paper we establish the normal and $\alpha$-stable approximations to the numbers of small cliques, cycles, and more general 2-connected subgraphs of a community affiliation graph.

We consider a diffusion process on the reals subject to the conditional probability that the process is positive from t = 0 to the present. We establish comparison results between the conditioned diffusion and a second unconditioned Markov diffusion. One result allows the initial process to be non-Markov before conditioning. A stronger comparison theorem is shown to hold in the Markov case.

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.

We study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U-statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U-statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

This note points out an error made in an earlier paper treating a diffusion equation for the probability distribution of gene frequencies under selection. The main interest lies in the orthogonality properties of certain eigenfunctions and the determination of the corresponding Fourier coefficients. Two particular cases of selection parameter values are considered. Approximate formulas are given for the two smallest eigenvalues for arbitrary values of the selection parameters.