Central limit theorem for weakly dependent variables

Mathematical Notes -     

Central limit theorem for stationary random fields

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 28, No. 4, pp. 758–769, October–December, 1988.     

Central limit theorem for shot noise fields

One investigates the asymptotic normality of integrals over unboundedly increasing sets of a random field, describing a shot noise in ^d ,d1. One shows how important here are roles played by the growth character of the considered sets and by the dimension d.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 28–36, 1989.     

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.     

Central limit theorem for stochastically continuous processes. Convergence to stable limit

LetX={X(t), t[0, 1]} be a stochastically continuous cadlag process. Assume that thek dimensional finite joint distributions ofX are in the domain of normal attraction of strictlyp-stable, 0<p<2, measure onR ^k for all 1k<. For functionsf, g such that _p (|X(x–X(u)|) >g(u–s) and _p (|X(s–X(t|)|X(t)–X(u|)>f(u–s), 0 s t u 1, conditions are found which imply that the distributions –(n ^–1/p (X ₁+···+X _n )),n1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX ₁,X ₂, ... are independent copies ofX and _p (Z)=sup _t<0 t ^pP{|Z|<t} denotes the weakpth moment of a random variable Z.     

Central limit theorem for nonlinear transforms of gaussian random vector fields

We prove a central limit theorem for integral functionals of nonlinear transforms of homogeneous, isotropic Gaussian random vector fields.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1057–1063, August, 1990.     

Central limit theorem for capacities

In this Note, our aim is to obtain the central limit theorem for capacities induced by sublinear expectations.     

Central limit theorem for Burgers equation

Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 88, No. 1, pp. 7–13, July, 1991.     

Central limit theorem for complex measures

We prove two central limit theorems for real identically distribution random variables where the distribution is a complex-valued Borel measure . The results involve the weak convergence of the suitably scaledn-fold convolution of certain complex atomic or absolutely continuous measures of spectral radius 1 to ahypergaussian measure whose Fourier-Stieltjes transform is exp(–² for a positive integer . The proof uses a generalization of the method of characteristic functions. Counter-examples are given to rather more general statements that had appeared previously in the literature. This research arose in connection with problems related to general tauberian theorems on the line for complexvalued summability methods which are discussed at the end of the paper. Some interesting examples are given generalizing well-known theorems related to Euler and Borel summability.     

A limit theorem for continuous selectors

Any (measurable) function K from Rⁿ to R defines an operator K acting on random variables X by K(X) = K(X₁,..., X_n), where the X_j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x₁, x₂,..., x_n) ∈ {x₁, x₂,..., x_n}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ω_H in the interval (0, 1) so that, for any random variable X, the iterates H^(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.     

Central limit theorem for traces of large random symmetric matrices with independent matrix elements

We study Wigner ensembles of symmetric random matricesA=(a _ij ),i, j=1,...,n with matrix elementsa _ij ,ij being independent symmetrically distributed random variables

Central limit theorem for constrained Poisson systems

We prove that for a class of constrained Poisson white noise fields, the scaling (continuum) limit exists and equals Gaussian white noise, indexed by mean zero test functions. Under natural conditions on the Lévy measure, the (Poisson) moments converge to their Gaussian counterparts.     

Central limit theorem for associated random variables

In this paper, we investigate an functional central limit theorem for a nonstatioaryd-parameter array of associated random variables applying the criterion of the tightness condition in Bickel and Wichura[1971]. Our results imply an extension to the nonstatioary case of invariance principle of Burton and Kim(1988) and analogous results for thed-dimensional associated random measure. These results are also applied to show a new functional central limit theorem for Poisson cluster random variables.     

Central limit theorem for asymmetric kernel functionals

Asymmetric kernels are quite useful for the estimation of density functions with bounded support. Gamma kernels are designed to handle density functions whose supports are bounded from one end only, whereas beta kernels are particularly convenient for the estimation of density functions with compact support. These asymmetric kernels are nonnegative and free of boundary bias. Moreover, their shape varies according to the location of the data point, thus also changing the amount of smoothing. This paper applies the central limit theorem for degenerate U-statistics to compute the limiting distribution of a class of asymmetric kernel functionals.     

Central limit theorem for finitely-dependent random variables

For sequences of finitely-dependent random variables, under rather general hypotheses we establish estimates of integrals of the form where F_n(x) is the distribution function of the normalized sum of random variables; (x) is the standard normal distribution function. In the proof we use relations obtained by the method of C. Stein. The results are applicable, in particular, to m-dependent random variables and fields.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 93–97, 1988.