Construction of non-Gaussian random fields with any given correlation structure

A positive definite function can be thought of as the covariance function of a Gaussian random field, according to the celebrated Kolmogorov existence theorem. A question of great theoretical and practical interest is: how could one construct a non-Gaussian random field with the given positive definite function as its covariance function? In this paper we demonstrate a novel and simple method for constructing many such non-Gaussian random fields, with the corresponding finite-dimensional distributions identified. Also, we show how to construct a non-Gaussian random field with a given negative definite function as its variogram.     

Extremal limit theorems for observations separated by random power law waiting times

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.     

A nonparametric repeated significance test with adaptive target sample size

In this article a general result is derived that, along with a functional central limit theorem for a sequence of statistics, can be employed in developing a nonparametric repeated significance test with adaptive target sample size. This method is used in deriving a repeated significance test with adaptive target sample size for the shift model. The repeated significance test is based on a functional central limit theorem for a sequence of partial sums of truncated observations. Based on numerical results presented in this article one can conclude that this nonparametric sequential test performs quite well.     

An application of the Bernoulli part to local limit theorems for moving averages on stationary sequences

We consider partial sums S_n of a general class of stationary sequences of integer-valued random variables, and we provide sufficient conditions for S_n to satisfy a local limit theorem. To prove this result, we introduce a concept called the Bernoulli part. The amount of Bernoulli part in S_n determines the extent to which the density of S_n is relatively flat. If in addition S_n satisfies a global central limit theorem, the local limit theorem follows.     

Variance estimation in the central limit theorem for Markov chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent and asymptotically normal. The main part of our analysis consists in presenting closed matrix forms for this new variance. Additionally, we prove the asymptotic equivalence between the empirical and the maximum likelihood estimation (MLE) for the stationary distribution.     

Almost sure central limit theorem for the maxima and sums of stationary Gaussian sequences

In this paper, we proved an almost sure central limit theorem for the maxima (after centered at the sample mean) and the partial sums of standardized stationary Gaussian sequences under some conditions related to the convergence rate of covariance functions, which extended the existing results.     

Extensions to Basu's theorem,factorizations, and infinite divisibility

We define a notion of approximate sufficiency and approximate ancillarity and show that such statistics are approximately independent pointwise under each value of the parameter. We do so without mentioning the somewhat nonintuitive concept of completeness, thus providing a more transparent version of Basu's theorem. Two total variation inequalities are given, which we call approximate Basu theorems.     

Geostochastic calculus

A stochastic calculus for a family of continuous measure-valued Markov processes is developed. Such processes arise naturally in the construction of stochastic models of spatially distributed populations. The stochastic calculus is a tool whereby a class of density-dependent models can be studied in terms of the multiplicative measure diffusion process. In this paper the stochastic integral is introduced in the space-time setting and a Cameron-Martin-Girsanov theorem is established.     

Strong invariance principles for triangular arrays of weakly dependent random variables

We establish a strong invariance principle for triangular arrays of a broad class of weakly dependent real random variables. We approximate the original array of dependent random variables by an array of rowwise independent standard normal variables. We demonstrate the functional central limit theorem and law of the iterated logarithm for the approximating array and thereby extend these results to the original array. Among several examples, we look at arrays used in describing the rate of convergence of estimators in regression analysis.     

Strassen type limit points for moving averages of a wiener process

Let {W(s); 8 ≥ 0} be a standard Wiener process, and let β_N = (2a_N (log (N/a_N) + log log N)^-1/2, 0 < α_N ≤ N < ∞, where α_N↑ and α_N/N is a non-increasing function of N, and define γ_N(t) = β_N[W(n_N + ta_N) ? W(n_N)), 0 ≥ t ≥ 1, with n_N = N – a_N. Let K = {x ? C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γ_N(t), t ? [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With a_N = N, our result reduces to Strassen's well-known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.     

Normal approximation to the hypergeometric distribution in nonstandard cases and a sub-Gaussian Berry–Esseen theorem

In this paper, we consider simple random sampling without replacement from a dichotomous finite population. We investigate accuracy of the Normal approximation to the Hypergeometric probabilities for a wide range of parameter values, including the nonstandard cases where the sampling fraction tends to one and where the proportion of the objects of interest in the population tends to the boundary values, zero and one. We establish a non-uniform Berry–Esseen theorem for the Hypergeometric distribution which shows that in the nonstandard cases, the rate of Normal approximation to the Hypergeometric distribution can be considerably slower than the rate of Normal approximation to the Binomial distribution. We also report results from a moderately large numerical study and provide some guidelines for using the Normal approximation to the Hypergeometric distribution in finite samples.     

Edgeworth expansion for signed linear rank statistics with regression constants

An Edgeworth expansion with remainder o(N^?1) is obtained for signed linear rank statistics under suitable assumptions. The theorem is proved for a wide class of score generating functions including the Chi-quantile function by adapting van Zwet's methodand Does's conditioning arguments.     

Multivariate Matsumoto–Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2M-X$2M-X$ theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Weso?owski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].     

Berry-Esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles

Berry-Esseen bounds of order O(n^−1/2) have been obtained for several classes of statistics. In this paper, the rates of convergence in central limit theorem for conditional empirical functions and conditional sample quantiles based on kernel estimators are studied for both conditional and unconditional distributions.     

A note on replacement policy comparisons from NBUC lifetime of the unit

Age and block replacement policies are commonly used in order to reduce the number of in-service failures when the systems are functioning indefinitely. In reliability theory, the lifetime of a system can be modeled by means of the NBUC aging class that is characterized throughout comparisons of the residual lives in the sense of the icx order. The purpose of this paper is to establish stochastic comparisons between the age (block) replacement policy and a renewal process with no planned replacements when the lifetime of the unit is NBUC. Supported by Ministerio de Ciencia y Tecnología under grant BFM2000-0362     

Limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesàro alpha-integrability assumption

Asymptotically negative association is a special dependence structure. By relating such dependence condition to residual Cesàro alpha-integrability and to strongly residual Cesàro alpha-integrability, some L_p-convergence and complete convergence results of the maximum of the partial sum are derived, respectively. In addition, some of these conclusions are based on a new Rosenthal type inequality concerning asymptotically negatively associated random variables, which is of independent interest.     

Occupancy numbers for dynamic heterogeneous populations: Estimate of particles lifetimes

A dynamic model of a heterogeneous population is studied. Particles belonging to a population are divided, at every time t, into a finite number of classes according to their types and the partition changes over time. The role of the occupancy numbers, namely the cardinality of each class, is highlighted. The relationship between the stochastic process of occupancy numbers and the process of particle types is analyzed. The main goal of this paper is the estimation of the lifetime of each particle at a given time t, when the observed data are the history of the process of the number of dead particles up to t. Furthermore, a discrete time approximation of the filter is given.