Esseen Inequality Research Articles

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z2 with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.

Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

For any 0 n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.

Precise Rates in the Law of the Iterated Logarithm for Associated Random Variables

Let {X i , i ≥ 1} be a sequence of strictly stationary and associated random variables with mean zero and finite variance. Set , M n = max 1≤k≤n |S k | and . By the maximal and moment inequalities and a Berry–Esseen inequality for associated random variables, we give the precise rates of a kind of weighted infinite series of , and as ϵ↘0, respectively, which reflect the convergence rates in the law of the iterated logarithm.

On the Upper Bound for the Absolute Constant in the Berry–Esseen Inequality

This paper describes the history of the search for unconditional and conditional upper bounds of the absolute constant in the Berry–Esseen inequality for sums of independent identically distributed random variables. Computational procedures are described. New estimates are presented from which it follows that the absolute constant in the classical Berry–Esseen inequality does not exceed 0.5129.

An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F n ,Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β3+0.429)≤0.33477(1+0.429)β3<0.4784β3 by virtue of the condition β3≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequality. As corollaries, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.

Berry–Esseen inequalities for discretely observed diffusions

For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, the paper compares the bounds on the rates of convergence to normality (Berry–Esseen type) of two approximate maximum likelihood estimators of the drift parameter based on the Itô and the Fisk–Stratonovich approximations of the continuous likelihood, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval. It shows that the Fisk–Stratonovich approximations performs better than the Itô approximations in the sense of having smaller variance.

On the accuracy of the normal approximation to the distributions of Poisson random sums

AbstractThe analogue of the Berry–Esseen inequality for Poisson random sums is improved in 4–11 times depending on the maximal order 2 &lt; s ≤ 3 of the finite absolute moment of the distribution of random summands. For the case, when s = 3, under additional assumption concerning the smoothness of distribution of summands we also obtained in some sense unimprovable estimate consisting of two summands: the first summand being the noncentral Lyapunov fraction with the coefficient equal to the unimprovable asymptotically exact constant and the second one decreases faster. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Berry–Esseen for Free Random Variables

An analogue of the Berry–Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n −1/2, the same as in the classical case. An example with binomial measures shows that this estimate cannot be improved without imposing further restrictions on convolved measures.

Sharpening of the Upper Bound of the Absolute Constant in the Berry–Esseen Inequality

The upper bound of the absolute constant in the classical Berry–Esseen inequality for sums of independent identically distributed random variables with finite third moments is lowered to $C\leqslant 0.7056$.

Berry–Esseen inequality for linear processes in Hilbert spaces

Under mild conditions we obtain the usual Berry–Esseen inequality for linear processes in Hilbert spaces.

Another Esseen-type inequality for multivariate probability density functions

An upper bound for the supremum of the absolute value of the difference of two multivariate probability density functions is obtained. The upper bound involves integrals of the absolute value of suitable transforms of the characteristic functions of the probability density functions. Results are similar to the work of Gamkrelidze (Theory Probab. Appl. 22 (1977) 877–880) on the Esseen's inequality for multidimensional distribution functions.

A New Asymptotic Expansion and Asymptotically Best Constants in Lyapunov's Theorem. II

A new asymptotic expansion is obtained in Lyapunov's central limit theorem for distribution functions of centered and normed sums of independent random variables which are not necessary identically distributed. It is applied to determine the asymptotically best constants in the Berry--Esseen inequality, thus solving problems about their optimal values raised by Kolmogorov and Zolotarev.

A New Asymptotic Expansion and Asymptotically Best Constants in Lyapunov's Theorm. I.

A new asymptotic expansion is obtained in Lyapunov's central limit theoremfor distribution functions of centered and normed sums of independent random variables which are not necessarily identically distributed. It is applied to determine the asymptotically best constants in the Berry--Esseen inequality, thus solving problems about their optimal values raised by Kolmogorov and Zolotarev.

On the Error-Bound in the Nonuniform Version of Esseen's Inequality in the L p -Metric

The aim of this paper is to investigate the known nonuniform version of Esseen's Inequality in the L P -metric, p ≥ 1, to get a numerical bound for the appearing constant L, see (2.1). For a long time the results given by several authors constate the impossibility of (2.1) in the most interesting case δ = 1, because the effect was observed, where 2 + δ, 0 < δ ≤ 1, is the order of the assumed moments of the considered independent random variables X k , k = 1, 2, …, n. Again making use of the method of conjugated distributions, cp.,5,11,19,26 we improve the well-known technique to show in the most interesting case δ = 1 the finiteness of the absolute constant L and to prove where . In the case δ ∈(0,1) we only give the analytical structure of L but omit numerical calculations. Finally an example on normal approximation of sums of l 2-valued random elements demonstrates the application of the nonuniform mean central limit bounds obtained here.

On Analogues of Berry–Esseen Inequality for Truncated Linear Combinations of Order Statistics

On Analogues of Berry–Esseen Inequality for Truncated Linear Combinations of Order Statistics

On the analytical structure of the constant in the nonuniform version of the esseen inequality

Let X1,X2 be a sequence of independent random variable such that . It is shown that for all neN and all xeR where c1<31.935 In the general case only tha analytical structure of C1=C1 is given without numerical calculation

The berry–esseen inequalities derived from restricted convergence

Let X1, X2, … be i.i.d.r.v. and write (X1+…Xn−An)/Bn˜Fn, where Bn >0.AnER1, n≥1. It is known that solely one–sided asymptotic assumptions imposed on Fn imply Fn0. In the present note we show that stronger one–sided assumptions lead even to the existence of EX1 3 so that the BERRY-ESSEEN inequalities hold true.

A generalization of the Esseen inequality for the concentration function

One proves an inquality for the concentration function of the random variable, representing a generalization of Esseen's inequality.

Estimates for the Levy-Prokhorov distance in terms of characteristic functions and some of their applications

One gives estimates of the Levy-Prokhorov distance between distributions in Rk in terms of integrals of the differences of the characteristic functions and of their derivatives, similar to Esseen's inequality.

On the Accuracy of Normal Approximation of the Probability of Hitting a Ball

On the Accuracy of Normal Approximation of the Probability of Hitting a Ball