A central limit theorem for a weighted power variation of a Gaussian process*

Let ρ be a real-valued function on [0, T], and let LSI(ρ) be a class of Gaussian processes over time interval [0, T], which need not have stationary increments but their incremental variance σ(s, t) is close to the values ρ(|t ? s|) as t → s uniformly in s ∈ (0, T]. For a Gaussian processesGfrom LSI(ρ), we consider a power variation V _n corresponding to a regular partition π _n of [0, T] and weighted by values of ρ(·). Under suitable hypotheses on G, we prove that a central limit theorem holds for V _n as the mesh of π _n approaches zero. The proof is based on a general central limit theorem for random variables that admit a Wiener chaos representation. The present result extends the central limit theorem for a power variation of a class of Gaussian processes with stationary increments and for bifractional and subfractional Gaussian processes.     

A central limit theorem for Hermitian polynomials of independent Gaussian variables

The conditions of asymptotic normality of the variables are studied for n and m, with H(x₁, ..., x_r) denoting Hermitian polynomials in (R^m)^r, and the ₁, ..., _n being independent Gaussian vectors in X=R^m with a zero mean and a unit correlation operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1681–1686, December, 1990.     

The central limit theorem and chaos

Let X be a compact metric space studies some relationships between stochastic and f ： X→ X be a continuous map. This paper and topological properties of dynamical systems. It is shown that if f satisfies the central limit theorem, then f is topologically ergodic and / is sensitively dependent on initial conditions if and only if / is neither minimal nor equicontinuous.     

A non-commutative central limit theorem

This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 123)     

Extreme values of a portfolio of Gaussian processes and a trend

We consider the extreme values of a portfolio of independent continuous Gaussian processes ( ) which are asymptotically locally stationary, with expectations and variances , and a trend for some constants with . We derive the probability for , which may be interpreted as ruin probability. AMS 2000 Subject Classifications Primary—60G15, 62G32, 91B28     

Fisher information and the central limit theorem

An Edgeworth-type expansion is established for the relative Fisher information distance to the class of normal distributions of sums of i.i.d. random variables, satisfying moment conditions. The validity of the central limit theorem is studied via properties of the Fisher information along convolutions.     

A tracial quantum central limit theorem

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.

     

A quantum-mechanical functional central limit theorem

Continuing an earlier work [4], properties of canonical Wiener processes are investigated. An analog of the sample path continuity property is obtained. A noncommutative counterpart of weak convergence is formulated. Operator processes (P_n, Q_n) analogous to the random-walk approximating processes of the Donsker invariance principle are defined in terms of a sequence (p_i, q_i) of pairs of quantum mechanical canonical observables satisfying hypotheses analogous to those of the classical central limit theorem. It is shown that P_n, Q_n) converges weakly to a canonical Wiener process.     

Central limit theorem for functionals of a generalized self-similar Gaussian process

We consider a class of self-similar, continuous Gaussian processes that do not necessarily have stationary increments. We prove a version of the Breuer–Major theorem for this class, that is, subject to conditions on the covariance function, a generic functional of the process increments converges in law to a Gaussian random variable. The proof is based on the Fourth Moment Theorem. We give examples of five non-stationary processes that satisfy these conditions.     

Fisher information inequalities and the central limit theorem

We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L² spaces and Poincaré inequalities, to provide a better understanding of the decrease in Fisher information implied by results of Barron and Brown. We show that if the standardized Fisher information ever becomes finite then it converges to zero.OTJ is a Fellow of Christs College, Cambridge, who helped support two trips to Yale University during which this paper was written.Mathematics Subject Classification (2000):Primary: 62B10 Secondary: 60F05, 94A17     

A central limit theorem for estimation in Gaussian stationary time series observed at unequally spaced times

The central limit theorem is proved for estimates of parameters which specify the covariance structure of a zero mean, stationary, Gaussian, discrete time series observed at unequally spaced times. The estimates considered are obtained by a single iteration from consistent estimates. The result also applies to the maximum likelihood estimate if it is consistent although consistency is not proved here. The essential condition on the sampling times is that the finite sample information matrix, when divided by the sample size, has a limit which is nonsingular and has finite norm. Some examples are presented to illustrate this condition.     

A central limit theorem for integer partitions

Recently, Hwang proved a central limit theorem for restricted Λ-partitions, where Λ can be any nondecreasing sequence of integers tending to infinity that satisfies certain technical conditions. In particular, one of these conditions is that the associated Dirichlet series has only a single pole on the abscissa of convergence. In the present paper, we show that this condition can be relaxed, and provide some natural examples that arise from the study of integers with restrictions on their digital (base-b) expansion.