A non-commutative central limit theorem

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

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 central limit theorem for the U-statistic

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 2, pp. 269–271, February, 1989.     

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.

     

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 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.     

A homoclinic version of the central limit theorem

The definitions of homoclinic partitions and transformations are given in situations that are standard for topological dynamics and ergodic theory. A variant of the central limit theorem is proved, the formulation of which makes use of homoclinic transformations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 80–91, 1990.     

On the central limit theorem in Banach spaces

It is shown, with the use of a concentration inequality of LeCam, that associated with an infinitely divisible random variable with values in a separable Banach space there is a Lévy-Khintchine formula. A partial converse of this fact is also proved. Relations between the continuity of the compound Poisson and the Gaussian variables associated with a Lévy measure are studied. A central limit theorem is obtained and examples are given.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

On the functional central limit theorem for martingales

Necessary and sufficient conditions for the functional central limit theorem for a double array of random variables are sought. It is argued that this is a martingale problem only if the variables truncated at some fixed point c are asymptotically a martingale difference array. Under this hypothesis, necessary and sufficient conditions for convergence in distribution to a Brownian motion are obtained when the normalization is given (i) by the sums of squares of the variables, (ii) by the conditional variances and (iii) by the variances. The results are proved by comparing the various normalizations with a natural normalization.Research sponsored in part by the Office of Naval Research, Contract N00014-75-C-0809     

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     

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

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On the central limit theorem in R k

Summary Let F ^*n denote the n th convolution of a distribution function F on R ^k and suppose that F has zero moments of the first order and finite second order moment matrix. It is well-known that F ^*n (n·) converges to a Gaussian d.f. as n + t8. These d.f.s determine measures F ^*n (nA) and (A) for Borelsets A, We present a method that admits the estimation of the remainder-term F ^*n (n A)- (A) when A belongs to a certain class of Borelsets. This class contains all convex sets. If F has finite absolute third order moments then the remainder-term is of the order n ^–1/2. Also the remainder term's dependence on the dimension k is given. These results strengthen and generalize earlier results in the same direction.This paper was first communicated at the Scandinavian mathematical congress in Oslo, August 1968.