On statistical properties of the lyapunov exponent of the generalized skew tent map

The statistical properties of the Lyapunov exponent of the chaotic generalized skew tent map is studied. Expressions of the mean and the variance of this Lyapunov exponent at each discrete time index are obtained. A sufficient condition for weakly mixing of the chaotic generalized skew tent map is derived, and the asymptotic distribution of its Lyapunov exponent is provided.     

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 and chaoticity

This paper studies the relations between stochastic properties and chaotic properties of a dynamical system satisfying the central limit theorem. For such a system, it is proved that every nonempty open set in its defining space contains a point with positive lower density of its return time set and that the system is syndetically sensitive, provided that it is strongly topologically ergodic. Moreover, it is shown that the system admits many chaotic properties if its domain is restricted to a tree.     

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.     

Central limit theorem and increment conditions

Let φ be a convex function defined on R⁺, with φ(0) = 0 and lim_x→0φ(x)/x=0. We show that there exists a uniformly bounded process (X_t) on [0,1] with continuous sample paths that satisfies the increment condition for every u < t, E(φ(| X_t− X_u|)) ⩽ t − u. but that fails the CLT.     

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 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 weakly dependent variables

Mathematical Notes -     

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.     

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 and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

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 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.