Asymptotic Properties of Gaussian Processes
01 Apr 1972-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 43, Iss: 2, pp 580-596
TL;DR: In this article, the authors considered separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho( t, s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or < 2.
Abstract: : The authors consider two problems for separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho(t,s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or <2. In showing the existence of such (stationary) processes for 0 = or < alpha < 2, the authors relate the magnitude of the tails of the spectral distributionsto the behavior of the covariance function at the origin. For 0 < alpha = or < 2, the authors obtain the asymptotic distribution of the maximum of X(t). This second result is used to obtain a result for X(t) as t approaches infinity similar to the 'so called' law of the iterated logarithm. (Author)
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01 Jan 2011
421 citations
TL;DR: In this article, the authors provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes.
Abstract: : The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes. The theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts. Emphasis is placed on stationary cases but other important classes (e.g. Mark of sequences) are included. Significant ideas and methods are described rather than details, and in particular the nature and role of important underlying point processes (such as exceedances and upcrossings) are emphasized. Applications are given to particular classes of process (e.g. normal, moving average) and connections with related theory (such as convergence of sums) are indicated.
270 citations
01 Jan 2011
TL;DR: In this paper, the authors consider density estimates of the usual type generated by a weight function and obtain limit theorems for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity.
Abstract: We consider density estimates of the usual type generated by a weight function. Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity. Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved.
204 citations
TL;DR: In this paper, the authors investigate the asymptotics for a Gaussian process with a variance function that is strictly increasing, regularly varying at infinity with index β > H, and vanishing at the origin.
106 citations
Cites background or methods from "Asymptotic Properties of Gaussian P..."
...It was soon realized that |s− t|α can be replaced by a regularly varying function (at zero) with minimal additional effort [37]; see also [3, 11, 23], to mention a few recent contributions....
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...This method has been applied sucessfully to find the asymptotics of P (supt∈[0,T ] X(t) > u), where X is either a stationary Gaussian process [33, 37] or a Gaussian process with a unique point of maximum variance [36]....
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TL;DR: Asymptotic confidence bands in non-parametric regression are constructed based on an undersmoothed local polynomial estimator and certain rates are derived for the error in coverage probability, which improves on existing results for methods that rely on the asymPTotic distribution of the maximum of some Gaussian process.
Abstract: In the present paper we construct asymptotic confidence bands in non-parametric regression. Our assumptions cover unequal variances of the observations and nonuni-form, possibly considerably clustered design. The confidence band is based on an undersmoothed local polynomial estimator. An appropriate quantile is obtained via the wild bootstrap. We derive certain rates (in the sample size n) for the error in coverage probability, which improves on existing results for methods that rely on the asymptotic distribution of the maximum of some Gaussian process. We propose a practicable rule for a data-dependent choice of the band-width. A small simulation study illustrates the possible gains by our approach over alternative frequently used methods.
104 citations
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TL;DR: In this paper, the authors considered the probability that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T. Several strict upper and lower bounds for P were given, along with some comparison theorems that relate P's for different covariance functions.
Abstract: This paper is concerned with the probability, P[T, r(τ)], that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T. Several strict upper and lower bounds for P are given, along with some comparison theorems that relate P's for different covariance functions. Similar results are given for F[T, r(τ)], the probability distribution for the interval between two successive zeros of the process.
786 citations
134 citations
TL;DR: In this article, the uniform Hdlder condition for real-valued, separable Gaussian processes with stationary increments was obtained for convex functions f(h) and f(m) for which the following events have probability 1:
Abstract: Introduction. H6lder conditions, both uniform and local, will be obtained for a wide class of separable, real-valued Gaussian processes with stationary increments. Let X(t) be such a process with E{X(t)} = 0 and E{(X(t))2} 0. The major portion of our results apply to processes for which a2(h) is concave for h E [0, 8]. This study is motivated by the well-known results for Brownian motion, the law of the iterated logarithm and Paul Levy's uniform Hdlder condition. Therefore, representing a real-valued, separable Gaussian process with stationary increments by the corresponding function a2(h), we seek functions f(h) and f(h) for which the following events have probability 1:
56 citations
TL;DR: A result is presented which generalizes the paper of Watanabe (Trans. AMS 148, 233-248) and is extended also to the nonstationary process treated by Watanbe.
Abstract: : The paper presents a result which generalizes the paper of Watanabe (Trans. AMS 148, 233-248). The result is extended also to the nonstationary process treated by Watanabe.
38 citations
33 citations