Central Limit Theorem Started at a Point for?Stationary Processes and Additive Functionals of?Reversible Markov Chains

In this paper we study the almost sure central limit theorem started at a point for additive functionals of a stationary and ergodic Markov chain via a martingale approximation in the almost sure sense. Some of the results provide sufficient conditions for general stationary sequences. We use these results to study the quenched CLT for additive functionals of reversible Markov chains.     

Law of large numbers and central limit theorem for randomly forced PDE's

We consider a class of dissipative PDE's perturbed by an external random force. Under the condition that the distribution of perturbation is sufficiently non-degenerate, a strong law of large numbers (SLLN) and a central limit theorem (CLT) for solutions are established and the corresponding rates of convergence are estimated. It is also shown that the estimates obtained are close to being optimal. The proofs are based on the property of exponential mixing for the problem in question and some abstract SLLN and CLT for mixing-type Markov processes.     

Some optimal bounds in the central limit theorem using zero biasing

The convergence rate in the central limit theorem (CLT) is investigated in terms of a wide class of probability metrics. Namely, optimal estimates for the proximity between a probability distribution and its zero bias transformation are derived. These new inequalities allow one to establish optimal rates of convergence in the CLT for sums of independent random variables with finite moments of order s, s∈(2,3], in terms of ideal metrics introduced by V.M. Zolotarev.     

IDENTIFICATION OF MULTIVARIATE ARMA MODELS

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Estimation of nonbinary random response

The paper treats a new approach to reducing the dimension of factors which affect the non-binary response variable Y. This is relevant in analysis of a number of stochastic models, for instance, in biological and medical studies. The quality of Y estimation by means of a function in those factors is described by a specified error functional. It involves a penalty function to take into account the importance of the forecast for different response values. The joint distribution of factors and response variable is unknown. Thus it is quite natural to employ for statistical inference the estimates of the error functional constructed by prediction algorithm and cross-validation procedure. One of our main results provides the criterion of strong consistency of such estimates as the number of observations tends to infinity. Due to this result one can identify the significant factors. We introduce also the regularized versions of estimates and establish for them the central limit theorem (CLT). The statistical variant of our CLT permits to construct the approximate confidence intervals for unknown error functional.     

An extrapolation theorem for nonlinear approximation and its applications

We prove an extrapolation theorem for the nonlinear m-term approximation with respect to a system of functions satisfying very mild conditions. This theorem allows us to prove endpoint L^p-L^q estimates in nonlinear approximation. As a consequence, some known endpoint estimates can be deduced directly and some new estimates are also obtained. Finally, applications of these new estimates are given to spherical m-widths and m-term approximation of the weighted Besov classes.     

Pointwise Exponentially Weighted Estimates for Approximation on the Half-Line

We establish a theorem in the style of Timan-Gopengauz which provides pointwise estimates for the simultaneous approximation of a function and its derivatives in the space C[0, ∞), with error measured in an exponentially weighted norm.     

Extremal Values of L-Functions Associated with Newforms in the Half-Plane of Absolute Convergence

We prove estimates for extremal values of L-functions associated with newforms f in the half-plane of absolute convergence of their Dirichlet series expansion. The proof is based on an effective version of Kronecker's approximation theorem and estimates for the Fourier coefficients of the newform f.     

Estimates for the rate of strong approximation in Hilbert space

We obtain infinite-dimensional corollaries of our recent results. We show that the finite-dimensional results imply meaningful estimates for the accuracy of strong Gaussian approximation of sums of independent identically distributed Hilbert space-valued random vectors with finite power moments. We establish that the accuracy of approximation depends substantially on the decay rate of the sequence of eigenvalues of the covariance operator of the summands.     

Large and Moderate Deviations Principles and Central Limit Theorem for the Stochastic 3D Primitive Equations with Gradient-Dependent Noise

Journal of Theoretical Probability - We establish the large deviations principle (LDP), the moderate deviations principle (MDP), and an almost sure version of the central limit theorem (CLT) for...     

一类更广泛的Захаров方程组的有限元分析

其中n=n(x,i)为离子的扰动量(实函数,ε为场量(复函数)。该方程组具有一系列重要性质,如具有一维孤立子解,即Langmuir孤立子,它的形成、发展和相互作用不同于KDV方程的孤立子,因而引起人们的兴趣和关注.[2]研究了这个方程组的周期初值问     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

Global weak solutions for a modified two-component Camassa-Holm equation

We obtain the existence of global-in-time weak solutions for the Cauchy problem of a modified two-component Camassa-Holm equation. The global weak solution is obtained as a limit of viscous approximation. The key elements in our analysis are the Helly theorem and some a priori one-sided supernorm and space-time higher integrability estimates on the first-order derivatives of approximation solutions.     

Statistical Version of the Central Limit Theorem for Vector-Valued Random Fields

The classical central limit theorem due to Newman for real-valued strictly stationary associated random fields is generalized to strictly stationary quasi-associated vector-valued random fields comprising, in particular, positively or negatively associated fields with finite second moments. We also establish a version of the CLT with random matrix normalization which allows us to construct approximate confidence intervals for the unknown mean vector.     

The central limit theorem for the Smoluchovski coagulation model

The general model of coagulation is considered. For basic classes of unbounded coagulation kernels the central limit theorem (CLT) is obtained for the fluctuations around the dynamic law of large numbers (LLN) described by the Smoluchovski equation. A rather precise rate of convergence is given both for LLN and CLT.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Estimates for Correlation in Dynamical Systems: From Hölder Continuous Functions to General Observables

For many dynamical systems that are popular in applications, estimates are known for the decay of correlation in the case of Hölder continuous functions. In the present article, we suggest an approach that allows us to obtain estimates for correlation in dynamical systems in the case of arbitrary functions. This approach is based on approximation and estimates are obtained with the use of known estimates for Hölder continuous functions. We apply our approach to transitive Anosov diffeomorphisms and derive the central limit theorem for the characteristic functions of certain sets with boundary of zero measure.     

On 3-monotone approximation by piecewise polynomials

We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L₁ approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation.     

Bounds for a class of linear functionals with applications to Hermite interpolation

A general estimation theorem is given for a class of linear functionals on Sobolev spaces. The functionals considered are those which annihilate certain classes of polynomials. An interpolation scheme of Hermite type is defined inN-dimensions and the accuracy in approximation is bounded by means of the above mentioned theorem. In one and two dimensions our schemes reduce to the usual ones, however our estimates in two dimensions are new in that they involve only the pure partial derivatives.This research was supported in part by the National Science Foundation under grant number N.S.F.-G.P.-9467.