An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Global fluctuations in general Dyson’s Brownian motion

We consider a system of diffusing particles on the real line in a quadratic external potential and with a logarithmic interaction potential. The empirical measure process is known to converge weakly to a deterministic measure-valued process as the number of particles tends to infinity. Provided the initial fluctuations are small, the rescaled linear statistics of the empirical measure process converge in distribution to a Gaussian limit for sufficiently smooth test functions. For a large class of analytic test functions, we derive explicit general formulae for the mean and covariance in this central limit theorem by analyzing a partial differential equation characterizing the limiting fluctuations.     

Mean field analysis of neural networks: A central limit theorem

We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural network’s fluctuations around its mean-field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. The proof relies upon weak convergence methods from stochastic analysis. In particular, we prove relative compactness for the sequence of processes and uniqueness of the limiting process in a suitable Sobolev space.     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

Fluctuations of shapes of large areas under paths of random walks

Summary We discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.     

Inference for 2-D GARCH models

The purpose of this paper is, in the first step, to consider a class of GMM estimators with interesting asymptotic properties and a reasonable number of computations for two dimensionally indexed Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. In the second step, we use the central limit theorem of Huang (1992) for spatial martingale differences to establish the LAN property for general two-dimensional discrete models on a regular grid with Gaussian errors. We then apply this result to the spatial GARCH model and derive the limit distribution of the maximum likelihood estimators of the parameters. Results of numerical simulations are presented.     

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.     

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Summary This paper applies the stochastic calculus of multiple Wiener-Itô integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Itô integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.Research supported by Office of Naval Research contracts N00014-86-K-0007 and N00014-89-J-1051     

Functional central limit theorem for super α-stable processes

A functional central limit theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α < d < 2α (0 < α ≤ 2), the limiting process is a Gaussian process, whose covariance is specified; for the critical dimension d= 2α and higher dimensions d < 2α, the limiting process is Brownian motion. Zhang Mei, Functional central limit theorem for the super-brownian motion with super-Brownian immigration, J. Theoret. Probab., to appear.     

Central limit theorems for realized volatility under hitting times of an irregular grid

We consider a continuous semi-martingale sampled at hitting times of an irregular grid. The goal of this work is to analyze the asymptotic behavior of the realized volatility under this rather natural observation scheme. This framework strongly differs from the well understood situations when the sampling times are deterministic or when the grid is regular. Indeed, neither Gaussian approximations nor symmetry properties can be used. In this setting, as the distance between two consecutive barriers tends to zero, we establish central limit theorems for the normalized error of the realized volatility. In particular, we show that there is no bias in the limiting process.     

随机环境中带移民的上临界分枝过程的Berry-Esseen界

考虑独立同分布的随机环境中带移民的上临界分枝过程(Zn).应用(Zn)与随机环境中不带移民分枝过程的联系,以及与相应随机游动的联系,在一些适当的矩条件下,本文证明关于log Zn的中心极限定理的Berry-Esseen界.     

On local behaviour of the phase separation line in the 2D Ising model

Summary. The aim of this note is to discuss some statistical properties of the phase separation line in the 2D low-temperature Ising model. We prove the functional central limit theorem for the probability distributions describing fluctuations of the phase boundary in the direction orthogonal to its orientation. The limiting Gaussian measure corresponds to a scaled Brownian bridge with direction dependent parameters. Up to the temperature factor, the variances of local increments of this limiting process are inversely proportional to the stiffness. Received: 6 June 1997/In revised form: 20 August 1997     

The Empirical Process for Bivariate Sequences with Long Memory

We establish a functional central limit theorem for the empirical process of bivariate stationary long range dependent sequences under Gaussian subordination conditions. The proof is based upon a convergence result for cross-products of Hermite polynomials and a multivariate uniform reduction principle, as in Dehling and Taqqu [Ann. Statist. 17 (1989), 1767–1783] for the univariate case. The effect of estimated parameters is also discussed.     

A central limit theorem for exchangeable random variables on a locally compact abelian group

A central limit theorem is given for uniformly infinitesimal triangular arrays of random variables in which the random variables in each row are exchangeable and take values in a locally compact second countable abelian group. The limiting distribution in the theorem is Gaussian.     

SOME CENTRAL LIMIT THEOREMS FOR SUPER BROWNIAN MOTION

1IntroductionLimittheoremsconstituteanimportantpartofthebranchingprocesstheory.Itisalwaysinterestingtofindconditionsunderwhichanon-degeneratelimitlawexists.SinceGaltonWatsonprocessesareunstable,peoplehavederivedlimittheoremsforthemthroughdevicessucllasmodifyingfactors,conditioning,immigration,etc.AunifiedtreatmentofthelimittheoryofGallon--WatsonprocessesisgiveninAthreyaandNey(1972).Someoftheabovementionedtechniqueshavealsobeenusedinthemeasure-valuedsettingtogetlimittheoremsforDawson-Watana…     

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

A univariate Hawkes process is a simple point process that is self-exciting and has a clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes processes have wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Asymptotic behaviors for functionals of random dynamical systems

In this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.     

Functional limit theorems for generalized quadratic variations of Gaussian processes

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).     

Lorenz Curve for Truncated and Censored Data

In this paper, we consider a nonparametric estimator of the Lorenz curve and Gini index when the data are subjected to random left truncation and right censorship. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. A law of the iterated logarithm for the Lorenz curve is also derived. Lastly, we obtain a central limit theorem for the corresponding Gini index.