On Beveridge-Nelson decomposition and limit theorems for linear random fields

We consider linear random fields and show how an analogue of the Beveridge-Nelson decomposition can be applied to prove limit theorems for sums of such fields.     

Random points associated with rectangles

The study of the distribution and moments of the distance between random points within a rectangle or in two coplanar rectangles is required in a wide variety of fields. Formulae for the distributions and arbitrary moments of the distance between two random points associated with one or two rectangles in various situations are given here explicitly. These explicit formulae will be helpful to those who work in various applied areas for the computations required in their problems. The third Author has partially been supported by C.N.R..     

Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.     

A Berry--Esseen Type Estimate for a Weakly Associated Vector Random Field

A Berry--Esseen type estimate is established for a weakly associated vector random field when sums are taken over regularly growing sets.     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

Invariance principles for non-isotropic long memory random fields

We prove that when a random field with bounded spectral density satisfies a Donsker type theorem, its dilated and properly normalised spectral field admits a weak limit. We apply this result to establish the convergence of partial sums for random fields obtained by filtering a white noise. In particular, we prove the convergence of partial sums for strongly-dependent fields whose memory does not satisfy the regularity conditions previously met in the literature.     

Conditional distribution of heavy tailed random variables on large deviations of their sum

It is known that large deviations of sums of subexponential random variables are most likely realised by deviations of a single random variable. In this article we give a detailed picture of how subexponential random variables are distributed when a large deviation of the sum is observed.     

On Toeplitz type quadratic functionals of stationary Gaussian processes

Summary A central limit theorem for Toeplitz type quadratic functionals of a stationary Gaussian processX(t),t, is proved, generalizing the result of Avram [1] for discrete time processes. The result is applied to the problem of nonparametric estimation of linear functionals of an unknown spectral density function. We give some upper bounds for the minimax mean square risk of the nonparametric estimators, similar to those by Ibragimov and Has'minskii [12] for a probability density function.     

Central limit theorems under weak dependence

This article is motivated by a central limit theorem of Ibragimov for strictly stationary random sequences satisfying a mixing condition based on maximal correlations. Here we show that the mixing condition can be weakened slightly, and construct a class of stationary random sequences covered by the new version of the theorem but not Ibragimov's original version. Ibragimov's theorem is also extended to triangular arrays of random variables, and this is applied to some kernel-type estimates of probability density.     

Conditioned rates of convergence in the CLT for sums and maximum sums

An interesting recent result of Landers and Roggé (1977, Ann. Probability5, 595–600) is investigated further. Rates of convergence in the conditioned central limit theorem are developed for partial sums and maximum partial sums, with positive mean and zero mean separately, of sequences of independent identically distributed random variables. As corollaries we obtain a conditioned central limit theorem for maximum partial sums both for positive and zero mean cases.     

On complete convergence for weighted sums of asymptotically linear negatively dependent random field

In this paper we establish the complete convergence for weighted sums of asymptotically linear negatively quadrant dependent random field, which contains a linear negatively quadrant dependent field and a ρ^∗${\rho }^{\ast }$-mixing random field.     

On the normal approximation for random fields via martingale methods

We prove a central limit theorem for strictly stationary random fields under a sharp projective condition. The assumption was introduced in the setting of random sequences by Maxwell and Woodroofe. Our approach is based on new results for triangular arrays of martingale differences, which have interest in themselves. We provide as applications new results for linear random fields and nonlinear random fields of Volterra-type.     

一类独立随机场的强定律的收敛率

§ 1　IntroductionDefinition1 .[1 ] A field{ Xi,i∈Nd} is called negatively associated(NA) if for every pair ofdisjoint subsets T1 ,T2 of Nd,Cov(f1 (Xi,i∈ T1 ) ,f2 (Xj,j∈ T2 ) )≤ 0 ,whenever f1 and f2 are coordinatewise increasing.Definition2 .[1 ] A field{ Xi,i∈Nd} is calledρ* -mixing ifρ* (s) =sup{ (ρ(S,T) ;S,T N,dist(S,T)≥ s}→ 0 (s→∞ ) ,whereρ(S,T) =sup{ |E(f -Ef) (g -Eg) |/‖ f -Ef‖2 ‖ g -Eg‖2 ,f∈ L2 (σ(S) ) ,g∈ L2 (σ(T) ) } .Definition 3.[1 ] A field { Xi…     

A connection between supermodular ordering and positive/negative association

In this paper, we show that a vector of positively/negatively associated random variables is larger/smaller than the vector of their independent duplicates with respect to the supermodular order. In that way, we solve an open problem posed by Hu (Chinese J. Appl. Probab. Statist. 16 (2000) 133) refering to whether negative association implies negative superadditive dependence, and at the same time to an open problem stated in Müller and Stoyan (Comparison Methods for Stochastic Modes and Risks, Wiley, Chichester, 2002) whether association implies positive supermodular dependence. Therefore, some well-known results concerning sums and maximum partial sums of positively/negatively associated random variables are obtained as an immediate consequence. The aforementioned result can be exploited to give useful probability inequalities. Consequently, as an application we provide an improvement of the Kolmogorov-type inequality of Matula (Statist. Probab. Lett. 15 (1992) 209) for negatively associated random variables. Moreover, a Rosenthal-type inequality for associated random variables is presented.     

An Inductive Proof of the Berry-Esseen Theorem for Character Ratios

Bolthausen used a variation of Stein’s method to give an inductive proof of the Berry-Esseen theorem for sums of independent, identically distributed random variables. We modify this technique to prove a Berry-Esseen theorem for character ratios of a random representation of the symmetric group on transpositions. An analogous result is proved for Jack measure on partitions. Received March 11, 2005     

CLT for linear random fields with martingale increments

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, which can be regarded as a continuation of the above-mentioned paper, CLT for sums of linear random field are proved in the case where innovations form martingale differences on the plane (that can be defined in several ways). In both papers, the so-called Beveridge–Nelson decomposition is used.     

Semi-selfdecomposable distributions and a new class of limit theorems

Self-decomposable distributions are given as limits of normalized sums of independent random variables. We define semi-selfdecomposable distributions as limits of subsequences of normalized sums. More generally, we introduce a way of making a new class of limiting distributions derived from a class of distributions by taking the limits through subsequences of normalized sums, and define the class of semi-selfdecomposable distributions and a decreasing sequence of subclasses of it. We give two kinds of necessary and sufficient conditions for distributions belonging to those classes, one is in terms of the decomposability of random variables and another is in terms of Lévy measures. Received: 1 May 1997 / Revised version: 5 February 1998     

Central limit theorem for positively associated stationary random fields

Positively associated stationary random fields on d-dimensional integral lattice arise in various models of mathematical statistics, percolation theory, statistical physics, and reliability theory. In this paper, we shall be concerned with a field with covariance functions satisfying a more general condition than summability. A criterion for the validity of the central limit theorem (CLT) for partial sums of a field from this class is established. The sums are taken over an increasing nest of parallelepipeds or cubes. The well-known conjecture of Newman stated that for an associated stationary random field the above condition on the covariance function should force the CLT to hold. As was shown by N. Herrndorf and A. P. Shashkin, this conjecture fails already for d = 1. In the present paper, the uniform integrability of the squared partial sums is shown as being of key importance for the CLT to hold. Thus, an extension of Lewis’s theorem proved for a sequence of random variables is obtained. Also, it is indicated how to modify Newman’s conjecture for any d. A representation of variances of partial sums of a field by means of slowly varying functions of several arguments is used in an essential way.     

Higher dimensional Dedekind sums in finite fields

We introduce Dedekind sums of a new type defined over finite fields. These are similar to the higher dimensional Dedekind sums of Zagier. The main result is the reciprocity law for them.     

Estimation of intrinsic processes affected by additive fractal noise

Fractal Gaussian models have been widely used to represent the singular behavior of phenomena arising in different applied fields; for example, fractional Brownian motion and fractional Gaussian noise are considered as monofractal models in subsurface hydrology and geophysical studies Mandelbrot [The Fractal Geometry of Nature, Freeman Press, San Francisco, 1982 [13]]. In this paper, we address the problem of least-squares linear estimation of an intrinsic fractal input random field from the observation of an output random field affected by fractal noise (see Angulo et al. [Estimation and filtering of fractional generalised random fields, J. Austral. Math. Soc. A 69 (2000) 1-26 [2]], Ruiz-Medina et al. [Fractional generalized random fields on bounded domains, Stochastic Anal. Appl. 21 (2003a) 465-492], Ruiz-Medina et al. [Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields, J. Multivariate Anal. 85 (2003b) 192-216]. Conditions on the fractality order of the additive noise are studied to obtain a bounded inversion of the associated Wiener-Hopf equation. A stable solution is then obtained in terms of orthogonal bases of the reproducing kernel Hilbert spaces associated with the random fields involved. Such bases are constructed from orthonormal wavelet bases (see Angulo and Ruiz-Medina [Multiresolution approximation to the stochastic inverse problem, Adv. in Appl. Probab. 31 (1999) 1039-1057], Angulo et al. [Wavelet-based orthogonal expansions of fractional generalized random fields on bounded domains, Theoret. Probab. Math. Stat. (2004), in press]). A simulation study is carried out to illustrate the influence of the fractality orders of the output random field and the fractal additive noise on the stability of the solution derived.