Functional central limit theorem for the volume of excursion sets generated by associated random fields

We prove a functional central limit theorem for the volume of the excursion sets generated by a stationary and associated random field with smooth realizations.     

Gaussian approximation of local empirical processes indexed by functions

Summary. An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated. Received: 11 January 1995/In revised form: 12 July 1996     

Almost sure invariance principles via martingale approximation

In this paper, we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, the almost sure central limit theorem, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts.     

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).     

On the sector condition and homogenization of diffusions with a Gaussian drift

In this paper we present the functional central limit theorem for a class of Markov processes, whose L²-generator satisfies the so-called graded sector condition. We apply the result to obtain homogenization theorems for certain classes of diffusions with a random Gaussian drift. Additionally, we present a result concerning the regularity of the effective diffusivity tensor with respect to the parameters related to the statistics of the drift. The abstract central limit theorem, see Theorem 2.2, is obtained by applying the technique used in Sethuraman et al. (Comm. Pure Appl. Math. 53 (2000) 972) to the case of infinite particle systems.     

Bootstraps of sums of independent but not identically distributed stochastic processes

A central limit theorem is developed for sums of independent but not identically distributed stochastic processes multiplied by independent real random variables with mean zero. Weak convergence of the Hoffmann–Jørgensen–Dudley type, as described in van der Vaart and Wellner (Weak Convergence and Empirical Processes, Springer, New York, 1996), is utilized. These results allow Monte Carlo estimation of limiting probability measures obtained from application of Pollard's (Empirical Processes: Theory and Applications, IMS, Hayward, CA, 1990) functional central limit theorem for empirical processes. An application of this theory to the two-parameter Cox score process with staggered entry data is given for illustration. For this process, the proposed multiplier bootstrap appears to be the first successful method for estimating the associated limiting distribution. The results of this paper compliment previous bootstrap and multiplier central limit theorems for independent and identically distributed empirical processes.     

Empirical processes of multidimensional systems with multiple mixing properties

We establish a multivariate empirical process central limit theorem for stationary R^d-valued stochastic processes (X_i)_i≥1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling et al. (2009) [9] in the univariate case. As an important technical ingredient, we prove a 2pth moment bound for partial sums in multiple mixing systems.     

Recurrence and ergodicity of diffusions

This article attempts to lay a proper foundation for studying asymptotic properties of nonhomogeneous diffusions, extends earlier criteria for transience, recurrence, and positive recurrence, and provides sufficient conditions for the weak convergence of a shifted nonhomogeneous diffusion to a limiting stationary homogenous diffusion. A functional central limit theorem is proved for the class of positive recurrent homogeneous diffusions. Upper and lower functions for positive recurrent nonhomogeneous diffusions are also studied.     

A functional limit theorem for <Emphasis Type="Italic">η</Emphasis>-weakly dependent processes and its applications

We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH(∞), LARCH(∞), Volterra processes, satisfies this theorem. Moreover, it provides numerous applications as well for bounding the distance between the empirical mean and the Gaussian measure than for obtaining central limit theorem for sample moments and cumulants. C. José Rafael León—Partially supported by the program ECOS-NORD of Fonacit, Venezuela.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

Reflected backward stochastic differential equation with jumps and RCLL obstacle

In this paper we study one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process when the solution is forced to stay above a right continuous left limits obstacle. We prove existence and uniqueness of the solution by using a penalization method combined with a monotonic limit theorem.     

A central limit theorem for martingales and an application to branching processes

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but R^N, the space of all real sequences and the metric used leads to a non-separable metric space.The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics

The Chung–Smirnov law of the iterated logarithm and the Finkelstein functional law of the iterated logarithm for empirical processes are used to establish new results on the central limit theorem, the law of the iterated logarithm, and the strong law of large numbers for L-statistics with certain bounded and smooth weight functions. These results are used to obtain necessary and sufficient conditions for almost sure convergence and for convergence in distribution of some well-known L-statistics and U-statistics, including Gini's mean difference statistic. A law of the logarithm for weighted sums of order statistics is also presented.     

An empirical process interpretation of a model of species survival

We study a model of species survival recently proposed by Michael and Volkov. We interpret it as a variant of empirical processes, in which the sample size is random and when decreasing, samples of smallest numerical values are removed. Micheal and Volkov proved that the empirical distributions converge to the sample distribution conditioned not to be below a certain threshold. We prove a functional central limit theorem for the fluctuations. There exists a threshold above which the limit process is Gaussian with variance bounded below by a positive constant, while at the threshold it is half-Gaussian.     

Fluctuations of interacting Markov chain Monte Carlo methods

We present a multivariate central limit theorem for a general class of interacting Markov chain Monte Carlo algorithms used to solve nonlinear measure-valued equations. These algorithms generate stochastic processes which belong to the class of nonlinear Markov chains interacting with their empirical occupation measures. We develop an original theoretical analysis based on resolvent operators and semigroup techniques to analyze the fluctuations of their occupation measures around their limiting values.     

An empirical process central limit theorem for dependent non-identically distributed random variables

This paper establishes a central limit theorem (CLT) for empirical processes indexed by smooth functions. The underlying random variables may be temporally dependent and non-identically distributed. In particular, the CLT holds for near epoch dependent (i.e., functions of mixing processes) triangular arrays, which include strong mixing arrays, among others. The results apply to classes of functions that have series expansions. The proof of the CLT is particularly simple; no chaining argument is required. The results can be used to establish the asymptotic normality of semiparametric estimators in time series contexts. An example is provided.     

A Maximal Inequality and a Functional Central Limit Theorem for set-indexed empirical processes

For the tail probabilities of a general set-indexed empirical process in an arbitrary sample space a maximal inequality is derived. In the case that the class of sets by which the process is indexed possesses a total ordering, the application of our inequality yields an elementary proof for a functional central limit theorem without involving such advanced techniques as symmetrization, stratification, chaining or Gaussian domination. Analogously, the inequality leads to a weak uniform law of large numbers (including convergence rate).     

Remarks on t-transformations of measures and convolutions

A family of transformations of probability measures is constructed, and used to define transformations of convolutions. The relations between moments and cumulants of a measure and its transformation are presented. For transformed classical and free convolutions the central limit measures and the Poisson type limit measures are computed. Families of non-commutative random variables are constructed, which are associated to these central limit measures. They provide examples of “position operators” which act on the Interacting Fock Spaces.     

The class of tenable zero-balanced Pólya urns with an initially dominant subset of colors

In Kholfi and Mahmoud (2011) the class of tenable irreducible nondegenerate zero-balanced Pólya urn schemes is introduced and its asymptotic behavior in various phases is studied. In the absence of an initially dominant subset of colors, the counts of balls of all the colors satisfy multivariate central limit theorems. It is reported there that the case of an initially dominant subset of colors poses challenges requiring finer asymptotic analysis. In the present investigation we follow up on this. Indeed, we characterize noncritical cases with an initially dominant subset of colors in which not all ball counts satisfy one multivariate central limit theorem, but rather a subset of the ball counts satisfies a singular multivariate central limit theorem. The rest of the cases are critical, in which all the ball counts satisfy a multivariate central limit theorem, but under a different scaling. However, for these critical cases the Gaussian phases are delayed considerably.