Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables

Summary Limit theorems with a non-Gaussian limiting distribution have been obtained, under appropriate conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence by a number of people. The normalization has typically been n , with 1/2<<1 where n is the sample size. Here examples of limit theorems are given for quadratic functions with long range memory (not instantaneous) with a normalization n , 0<<1/2.Research supported in part by the Office of Naval Research     

Almost sure central limit theorems on the Wiener space

In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law to a normal distribution.     

Limit Theorems for the Non-linear Functional of Stationary Gaussian Processes

In this paper we consider two functional limit theorems for the non-linear functional of the stationary Gaussian process satisfying short range dependence conditions: the functional CLT for partial sum processes and the uniform CLT for a special class of functions. To carry out the proofs, we develop Rosenthal type inequalities for the functional of Gaussian processes.     

Remarks on limit theorems for nonlinear functionals of Gaussian sequences

Summary Limit theorems for sums of nonlinear functionals of Gaussian sequences typically obtain as limit distribution that of a single term in an expansion given by Dobrushin [1] for a process subordinate to a Gaussian process. Here we show how one can obtain limit theorems of this type where the limit distribution is that of a full expansion of Dobrushin's type.This research is supported in part by Office of Naval Research contract N00014-81-K-003 and National Science Foundation Grant No. DMS 83-12106     

On partial sums of a gaussian sequence with stationary increments

We establish large increment properties for a Gaussian sequence with stationary increments under global conditions. Limit theorems for partial sums of the sequence with nonpositive correlation functions or the correlation functions on their speed of convergence to zero are proved via estimating a probability inequality on the supremum of Gaussian processes     

Some examples of application of the metric entropy method

We show how the metric entropy method can be substituted for the dyadic chaining, to prove in a unified setting several classical results. Among them are Stechkin's theorem, Gál--Koksma theorems and quantitative Borel--Cantelli lemmas. We give simpler proofs and improve some of these results. Two classes of examples are given: firstly stationary Gaussian sequences with applications to upper and lower classes and the law of the iterated logarithm for subsequences, and secondly in Diophantine approximation relatively to Gál and Schmidt's theorems.     

非齐次可列马尔可夫过程的轨道性质

§1 状态分类 定义1.1 设I是非负整数集,P={P_(ij)(s,t)|i,j∈I,α≤s≤t≤b}是转移函数矩阵。称P对i在t右标准,若limp_(ii)(t,t+h)=1;称P对i在t左标准,若limP_(ii)(t-h,t)=1.若P对i在t同时为右标准的和左标准的,则称P对i在t标准。若P对i在每个t标准,则称P对i标准。P对i右标准或左标准与此类似。若P对每个i标准,则称P标准。P右标准或左标准与此类似(参看[5]、[6])。     

Steady-state solutions of the Vlasov-Maxwell system and their stability

The study of the Vlasov-Maxwell system is reduced to nonlinear elliptic equations in the stationary case and hyperbolic equations in the nonstationary case. On this basis, theorems on the existence of solutions and sufficient conditions of Lyapunov stability are obtained. The cases are considered when electromagnetic fields and distribution functions can be constructed in an implicit form.     

Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors

Summary This paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are Gaussian, or a function of Gaussian random variables, that are strictly stationary and long range dependent. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors. The latter result, in turn, yields the asymptotic distribution of the Jaeckel (1972) rank estimators. The paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent.Research of this author was partly supported by the NSF grant: DMS-9102041     

Comparison and nonoscillation results for perturbed nonlinear differential equations

Summary Nonoscillation theorems for perturbed second order nonlinear differential equations are obtained. A nonlinear Picone type identity is introduced to obtain some Sturm-Picone type comparison theorems for nonlinear equations. Entrata in Redazione il 19 gennaio 1977. Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.     

Asymptotic properties of LSE of regression coefficients on singular random fields observed on a sphere

We present some upper bounds on the rate of convergence in the central limit theorem for normalized least square estimates (LSE) in a spherical regression model with long range dependence (LRD) stationary errors. The used method is based on the asymptotic analysis of orthogonal expansion of non-linear functionals of homogeneous isotropic Gaussian random fields and on the Kolmogorov distance. The theory have many applications in science for instance in evaluating the COBE data.     

A mixture-type limit theorem for nonlinear functions of Gaussian sequences

We show that, for a certain class of nonlinear functions of Gaussian sequences, the limiting distribution of normalized sums of the nonlinear function values of a sequence is the convolution of a Gaussian distribution with another non-Gaussian distribution.     

The Empirical Process for Bivariate Sequences with Long Memory

We establish a functional central limit theorem for the empirical process of bivariate stationary long range dependent sequences under Gaussian subordination conditions. The proof is based upon a convergence result for cross-products of Hermite polynomials and a multivariate uniform reduction principle, as in Dehling and Taqqu [Ann. Statist. 17 (1989), 1767–1783] for the univariate case. The effect of estimated parameters is also discussed.     

A test for a hypothesis on the correlation function of Gaussian random processes

A test is proposed for a hypothesis on the correlation function of general Gaussian random processes. The test is based on theorems on estimates of the distribution of the supremum of sample estimators of correlation functions of Gaussian processes. For a wide class of stationary processes formulas are given that allow the test to be used immediately. Bibliography: 4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 61–74.     

Convergence theorems and Tauberian theorems for functions and sequences in Banach spaces and Banach lattices

We prove generalized convergence theorems and Tauberian theorems for vector-valued functions and sequences of growth order γ − 1 with γ > 0 and for positive functions and sequences in Banach lattices. Then the general results are applied to obtain some interesting particular Tauberian results for various examples of operator semigroups. Among them are mean ergodic theorems for Cesàro-mean-bounded semigroups (discrete and continuous) of operators and for semigroups of positive operators. Research supported in part by the National Science Council of Taiwan. Current address: 19-18, Higashi-hongo 2-chome, Midori-ku, 226-0002 Japan.     

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Summary This paper applies the stochastic calculus of multiple Wiener-Itô integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Itô integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.Research supported by Office of Naval Research contracts N00014-86-K-0007 and N00014-89-J-1051     

Optimal rates for parameter estimation of stationary Gaussian processes

We study rates of convergence in central limit theorems for partial sums of polynomial functionals of general stationary and asymptotically stationary Gaussian sequences, using tools from analysis on Wiener space. In the quadratic case, thanks to newly developed optimal tools, we derive sharp results, i.e. upper and lower bounds of the same order, where the convergence rates are given explicitly in the Wasserstein distance via an analysis of the functionals’ absolute third moments. These results are tailored to the question of parameter estimation, which introduces a need to control variance convergence rates. We apply our result to study drift parameter estimation problems for some stochastic differential equations driven by fractional Brownian motion with fixed-time-step observations.     

Asymptotic properties of correlation bounds in functional spaces. II

This paper is an extension of [11]. Starting from the results of our first paper we prove by inclusion theorems that bounds for the correlation function of a stationary Gaussian process in the space of continuous functions with weight are strongly consistent and asymptotically normal. We construct the simplest functional confidence intervals in these spaces for the indeterminate correlation function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 322–329, March, 1991.     

Asymptotic properties of correlation bounds in functional spaces. II

This paper is an extension of [11]. Starting from the results of our first paper we prove by inclusion theorems that bounds for the correlation function of a stationary Gaussian process in the space of continuous functions with weight are strongly consistent and asymptotically normal. We construct the simplest functional confidence intervals in these spaces for the indeterminate correlation function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 322–329, March, 1991.     

Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the hurst parameter

In this contribution, the statistical performance of the wavelet-based estimation procedure for the Hurst parameter is studied for non-Gaussian long-range dependent processes obtained from point transformations of Gaussian processes. The statistical properties of the wavelet coefficients and the estimation performance are compared both for processes having the same covariance but different marginal distributions and for processes having the same covariance and same marginal distributions but obtained from different point transformations, analyzed using mother wavelets with different number of vanishing moments. It is shown that the reduction of the dependence range from long to short by increasing the number of vanishing moments, observed for Gaussian processes, and at the origin of the popularity of the wavelet-based estimator, does not hold in general for non-Gaussian processes. Crucially, it is also observed that the Hermite rank of the point transformation impacts significantly the statistical properties of the wavelet coefficients and the estimation performance and also that processes having identical marginal distributions and covariance function can yet yield significantly different estimation performance. These results are interpreted in the light of central and noncentral limit theorems that are fundamental when dealing with long-range dependent processes. Moreover, it will be shown that, on condition that estimation is performed using a range of scales restricted to the coarsest practically available, an approximate, yet analytical and simple to use in practice, formula can be proposed for the evaluation of the variance of the wavelet-based estimator of the Hurst parameter.