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.     

On the central limit theorem for f (n k x)

By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) _n≥1 with 2π-periodic ${f\in L^2}$ this phenomenon is generally not valid and the asymptotic behavior of ( f (n _k x)) _k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n _k . By the classical theory, the central limit theorem holds for f (n _k x) if n _k  = 2 ^k , or if n _k+1/n _k → α with a transcendental α, but it fails e.g., for n _k  = 2 ^k  ? 1. The purpose of our paper is to give a necessary and sufficient condition for f (n _k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n _k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened “infinitesimally”.     

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.     

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     

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

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On the remained term for the central limit theorem

Summary An upper bound for the remainder term of the Edgeworth expansion for the distribution of the normalized sum of independent and identically distributed random variables is given in terms of 3rd and 4th order moments, together with the total variation of the probability density function of the underlying distribution. The Institute of Statistical mathematics     

On the central limit theorem for modulus trimmed sums

We prove a functional central limit theorem for modulus trimmed i.i.d. variables in the domain of attraction of a nonnormal stable law. In contrast to the corresponding result under ordinary trimming, our CLT contains a random centering factor which is inevitable in the nonsymmetric case. The proof is based on the weak convergence of a two-parameter process where one of the parameters is time and the second one is the fraction of truncation.     

On the central limit theorem for lacunary trigonometric series

Хорошо известно, что в ероятностное поведе ние лакунарного тригоно метрического ряда {cos 2πn _kx} тесно связ ано с «критическим» у словием лакунарности (*) $$\frac{{n_{k + 1} }}{{n_k }} \geqq 1 + \frac{{c_k }}{{\sqrt k }},c_k \to \infty $$ . Например, если выполн ено условие (*), то последовательность {cos2πn _kx} удовлетворяет центральной предель ной теореме, и при этом условие (*) не может быть ослабле но. Для последовательносте й, удовлетворяющих (*), и звестны и другие результаты по добного рода, в то время как для более медленно расту щих последовательносте й {n_k} не известно, по-видимому, ничего. В с татье развит метод, ко торый при помощи мартингально й техники позволяет проводить исследование систем {cos 2πn_kx} для последовательно стей, не удовлетворяю щих условию (*). Получено про стое объяснение условия (*), изучено, как «пропа-дает» центральная предель ная теорема при посте пенном ослаблении условия (*) и дока-заны некоторые центральн ые предельные теорем ы в отсутствие этого усл овия. Получены другие предельные те оремы для {cos 2πn_kx}, напри мер, закон повторного лог арифма и принципы инвариантн ости.     

On the global central limit theorem forM-dependent random variables

Published in Lietuvos Matematikos Rinkinys, Vol. 34, No. 2, pp. 259–265, April–June, 1994.     

On estimation of the remainder in the central limit theorem

A nonuniform estimate of the remainder in the central limit theorem is obtained for a sequence of independent, identically distributed random variables. This estimate is a generalization of an earlier result of L. V. Osipov and the author. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 142–146.     

On the universal A.S. central limit theorem

Let (X _k ) be a sequence of independent r.v.’s such that for some measurable functions gk : R ^k → R a weak limit theorem of the form

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.     

On the rate of convergence in the entropic central limit theorem

We study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition.Mathematics Subjects Classification (2000):94A17; 60F05Supported in part by the EU Grant HPMT-CT-2000-00037, The Minkowski center for Geometry and the Israel Science Foundation.Supported in part by NSF Grant DMS-9796221.Supported in part by EPSRC Grant GR/R37210.Supported in part by the BSF, Clore Foundation and EU Grant HPMT-CT-2000-00037.     

A non-commutative central limit theorem

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

On the real accuracy of approximation in the central limit theorem

We consider various approximations in the central limit theorem for distributions of sums of independent random variables. We study how many summands in the normalized sums guarantee the precision 10⁻³ for these approximations. It turns out that for the same distribution but different approximations this number varies from hundreds of thousands to a few tens.