Chang C. Y. Dorea

On functional central limit theorem for stationary martingale random fields

O. Introduction The term random field is often used to denote a collection of random variables with a parameter space which is a subset of the q-dimensional Euclidean space Rq. Stationary random fields are of great practical importance and hence also of theoretical interest. Examples of random fields occur in biological investigations concerning the distribution of plants or animals over a given area, when q=2 and t=(h, t2) is a point of the area. In problems involving propagation of electromaodaetic waves through random media the natural parameter space is a subset of R 4, representing space and time. Further important examples occur in the theory of turbulance where, for example, one may consider the case q=4 and t is a point in space-time, while r ~2(t), r are the velocity components of a turbulent fluid at the point t. Multiparameter stochastic process (the so-called random field) plays a prominent role in weak convergence of empirical process to Kiefer process (a two-dimensional Brownian bridge), Brownian sheets, and sample spacings. In this paper we extend the concept of martingale to random fields and obtain a functional central limit theorem for such random fields. An important example of martingales with a partially ordered parameter set is the following generalization of Wiener process. Let dq be the family of all Borel sets in Rq having finite Lebesgue measure. Let {X a, A ~d q} be a real Gaussian additive random set function with E(Xa) = 0, E(XaXB)=m (A A B) where m denotes the Lebesgue measure. Intuitively, Xa can be thought of as the integral over A of a Gaussian White noise. Such integral of Gaussian White noise has extensively been used by Physicists and engineers.

On Determination of the Order of a Markov Chain

In this paper we propose a new and more general criterion (the efficient determination criterion, EDC) for estimating the order of a Markov chain. The consistency and the strong consistency of the estimates have been established under mild conditions.

Multistage Markov Chain Modeling of the Genetic Algorithm and Convergence Results

The genetic algorithm (GA) has been widely used to solve combinatorial global optimization problems. Despite the successes that GA encounters in practical applications, there exist few precise results on its behavior. In this article, we formulate a fully rigorous mathematical modeling of GA as a multistage Markov chain and derive convergence results. Variations that include the simulated annealing algorithm and the GA with superindividual are considered.

Estimation of the parameters of a time series subject to the error of rotation sampling

A sequence of repeated survey estimates of a population mean Xt is approached as a time series subject to measurement error ut . In rotation sampling designs the error ut can be modeled as a moving average process. For an autoregressive process Xt we propose an asymptotically efficient estimation method for the parameters of Xt based upon the properties of the least squares estimators of the model Yt = Xt + ut . A Monte Carlo study examining the proposed estimation methods was conducted.

A note on goodness-of-fit statistics with asymptotically normal distributions

A generalization of the Cramer-vonMises L2 distance is proposed. It gives rise to a class of goodness-of-fit statistics that is difficult to analyze using traditional techniques based on empirical distributions but can easily be modified to yield null and non null limiting normal distributions. The family index may be used to maximize the power of the test for a specific alternative hypothesis. The procedure presented here is shown to work for Watsons modification for circular data and also when testing symmetry about the zero. The problem of testing two-samples is also presented. All procedures presented here are distributions-free and can be used equally for univariate or multivariate data.

Kernel estimation for stationary density of Markov chains with general state space

Let {Xn}n≥0 be a Markov chain with stationary distributionf(x)ν(dx), ν being a σ-finite measure onE⊂Rd. Under strict stationarity and mixing conditions we obtain the consistency and asymptotic normality for a general class of kernel estimates off(·). When the assumption of stationarity is dropped these results are extended to geometrically ergodic chains.

Asymptotic Distribution of Extremes of Randomly Indexed Random Variables

Let {Xn} be a sequence of i.i.d. random variables and let {τk} be a sequence of random indexes. We study the problem of the existence of non-degenerated asymptotic distribution for min{Xτ1,..., Xτn}.

Estimation of the extreme value and the extreme points

Letf be a continuous function defined on some domainA andX 1,X 2, ... be iid random variables. We estimate the extreme value off onA by studying the limiting distribution of min {f(X 1), ...,f(X n )} or max {f(X 1), ...,f(X n )} properly normalized. Sufficient conditions for the existence of the limiting distribution as well as a characterization of the limiting distribution relative to the extreme points off will be provided. A discussion of the multidimensional case is also carried out.

A note on the Lindeberg condition for convergence to stable laws in Mallows distance

We correct a condition in a result of Johnson and Samworth (Bernoulli 11 (2005) 829--845) concerning convergence to stable laws in Mallows distance. We also give an improved version of this result, setting it in the more familiar context of a Lindeberg-like condition.

Differentiability preserving properties of a class of semigroups

AbstractIt is well known that for a large class of Markov process the associated semi-group T(t)f(x)=∫f(y)P(t,x;dy) satisfies the Kolmogorov backward differential equation, that is, if u(t,x)=T(t)f(x) then