Limit Theorem For Random Fields Research Articles

Directional phantom distribution functions for stationary random fields

We give necessary and sufficient conditions for the existence of a phantom distribution function for a stationary random field on a regular lattice. We also introduce a less demanding notion of a directional phantom distribution, with potentially broader area of applicability. Such approach leads to sectorial limit properties, a phenomenon well-known in limit theorems for random fields. An example of a stationary Gaussian random field is provided showing that the two notions do not coincide. Criteria for the existence of the corresponding notions of the extremal index and the sectorial extremal index are also given.

A central limit theorem for functions of stationary max-stable random fields on [formula omitted

Max-stable random fields are very appropriate for the statistical modelling of spatial extremes. Hence, integrals of functions of max-stable random fields over a given region can play a key role in the assessment of the risk of natural disasters, meaning that it is relevant to improve our understanding of their probabilistic behaviour. For this purpose, in this paper, we propose a general central limit theorem for functions of stationary max-stable random fields on Rd. Then, we show that appropriate functions of the Brown–Resnick random field with a power variogram and of the Smith random field satisfy the central limit theorem. Another strong motivation for our work lies in the fact that central limit theorems for random fields on Rd have been barely considered in the literature. As an application, we briefly show the usefulness of our results in a risk assessment context.

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of $$\alpha $$ -mixing (for local statistics) and exponential $$\alpha $$ -mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.

Non-central limit theorems for random fields subordinated to gamma-correlated random fields

A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in dd-dimensional space. As a particular case, integrals of non-linear functions of chi-squared random fields, with Laguerre rank being equal to one and two, are studied. When the Laguerre rank is equal to one, the characteristic function of the limit random variable, given by a Rosenblatt-type distribution, is obtained. When the Laguerre rank is equal to two, a multiple Wiener–Ito stochastic integral representation of the limit distribution is derived and an infinite series representation, in terms of independent random variables, is obtained for the limit.

Spectral covariance and limit theorems for random fields with infinite variance

In the paper, we continue to investigate measures of dependence for random variables with infinite variance. For random variables with regularly varying tails, we introduce a general class of such measures, which includes the codifference and the spectral covariance. In particular, we investigate the α -spectral covariance, a new measure from this general class, for linear random fields with infinite second moment. Under some conditions on the filter of a linear random field, we investigate asymptotic properties of the α -spectral covariance for linear random fields with infinite variance. We also provide an application of spectral covariances for limit theorems for stationary and associated random fields with infinite variance.

A general result in almost sure central limit theory for random fields

In this note, we investigate, under some mild conditions, the almost sure central limit theorem for random fields with general weight sequences.

A limit theorem for random fields with a singularity in the spectrum

A limit theorem for random fields with a singularity in the spectrum

On limit theorems for random fields

A complete separable metric space of functions defined on the positive quadrant of the plane is constructed. The characteristic property of these functions is that at every point x there exist two lines intersecting at this point such that limits limy→x f (y) exist when y approaches x along any path not intersecting these lines. A criterion of compactness of subsets of this space is obtained.

Local Vote Decision Fusion for Target Detection in Wireless Sensor Networks

This paper examines the problem of target detection by a wireless sensor network. Sensors acquire measurements emitted from the target that are corrupted by noise, and initially make individual decisions about the presence/absence of the target. We propose the local vote decision fusion algorithm, in which sensors first correct their decisions using decisions of neighboring sensors, and then make a collective decision as a network. An explicit formula that approximates the system's decision threshold for a given false alarm rate is derived using limit theorems for random fields, which provides a theoretical performance guarantee for the algorithm. We examine both distance- and nearest-neighbor-based versions of the local vote algorithm for grid and random sensor deployments and show that, in many situations, for a fixed-system false alarm, the local vote correction achieves significantly higher target detection rate than decision fusion based on uncorrected decisions. The algorithm does not depend on the signal model and is shown to be robust to different types of signal decay. We also extend this framework to temporal fusion, where information becomes available over time.

Almost sure central limit theorems for random fields

AbstractA general almost sure limit theorem is presented for random fields. It is applied to obtain almost sure versions of some (functional) central limit theorems. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Central Limit Theorem for Random Fields of Negatively Associated Processes

A central limit theorem for negatively associated random fields is established under the fairly general conditions. We use the finite second moment condition instead of the finite (2+δ)th moment condition used by Roussas.(15) A similar result is also given for positively associated sequences.

Exponential inequalities and functional central limit theorems for random fields

We establish new exponential inequalities for partial sums of random fields. Next, using classical chaining arguments, we give sufficient conditions for partial sum processes indexed by large classes of sets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, the condition is expressed in terms of a series of conditional expectations. For non-uniform ϕ -mixing random fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.

Non-Uniform Bounds for the Error in the Central Limit Theorem for Random Fields Generated by Functions of Independent Random Variables

Non-Uniform Bounds for the Error in the Central Limit Theorem for Random Fields Generated by Functions of Independent Random Variables

An Approach to Proving Limit Theorems for Dependent Random Variables

Previous article Next article An Approach to Proving Limit Theorems for Dependent Random VariablesB. S. NakhapetyanB. S. Nakhapetyanhttps://doi.org/10.1137/1132080PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– 48:1287 0219.60027 Google Scholar[2] A. V. Bulinskii, Limit Theorems for Random Processes and Fields, Vol. Moscow, Moscow State University Press, 1981, (In Russian.) Google Scholar[3] Hiroshi Takahata, On the central limit theorem for weakly dependent random fields, Yokohama Math. J., 31 (1983), 67–77 85m:60052 0532.60043 Google Scholar[4] E. Bolthausen, On the central limit theorem for stationary mixing random fields, Ann. Probab., 10 (1982), 1047–1050 84c:60035 0496.60020 CrossrefGoogle Scholar[5] B. S. Nakhapetyan, Central limit theorem for a random field satisfying a strong mixing condition, Dokl. Armen. SSR, 61 (1075), 210–213, (In Russian.) Google Scholar[6] B. S. Nakhapetyan, The central limit theorem for random fields satisfying a strong mixing conditionMulticomponent Random Systems, Nauka, Moscow, 1978, 276–288, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fieldsThe Annals of Applied Probability, Vol. 32, No. 3 | 1 Jun 2022 Cross Ref Compound Poisson approximation for regularly varying fields with application to sequence alignmentBernoulli, Vol. 27, No. 2 | 1 May 2021 Cross Ref Conditional Hazard Estimate for Functional Random FieldsJournal of Statistical Theory and Practice, Vol. 8, No. 2 | 24 March 2014 Cross Ref Robust nonparametric estimation for spatial regressionJournal of Statistical Planning and Inference, Vol. 140, No. 7 | 1 Jul 2010 Cross Ref Asymptotic normality of frequency polygons for random fieldsJournal of Statistical Planning and Inference, Vol. 140, No. 2 | 1 Feb 2010 Cross Ref Estimation of the trend function for spatio-temporal modelsJournal of Nonparametric Statistics, Vol. 21, No. 5 | 1 Jul 2009 Cross Ref Kernel regression estimation for random fieldsJournal of Statistical Planning and Inference, Vol. 137, No. 3 | 1 Mar 2007 Cross Ref LOCAL LINEAR FITTING UNDER NEAR EPOCH DEPENDENCEEconometric Theory, Vol. 23, No. 01 | 6 December 2006 Cross Ref Local linear spatial regressionThe Annals of Statistics, Vol. 32, No. 6 | 1 Dec 2004 Cross Ref Stochastic Dynamics of Fluctuations in Classical Continuous SystemsJournal of Functional Analysis, Vol. 185, No. 1 | 1 Sep 2001 Cross Ref Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal DistributionLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Kernel density estimation for random fields (density estimation for random fields)Statistics & Probability Letters, Vol. 36, No. 2 | 1 Dec 1997 Cross Ref Kernel density estimation for random fields: The L1 TheoryJournal of Nonparametric Statistics, Vol. 6, No. 2-3 | 1 Jan 1996 Cross Ref Kernel density estimation on random fieldsJournal of Multivariate Analysis, Vol. 34, No. 1 | 1 Jul 1990 Cross Ref Volume 32, Issue 3| 1988Theory of Probability & Its Applications389-571 History Submitted:01 April 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132080Article page range:pp. 535-539ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Some limit theorems for random fields

We prove a central limit theorem with remainder and an iterated logarithm law for collections of mixing random variables indexed byZ d,d≧1. These results are applicable to certain Gibbs random fields.

On a limit theorem for random fields of the shot effect type

On a limit theorem for random fields of the shot effect type