We address the question of a Berry Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting operator-valued semicircular element.

Recursive and closed form upper bounds are o¤ered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The approximation error in the CLT obtained from these new bounds vanishes at a rate O(n ); provided that the common distribution of the summands po...

In this paper, we obtain an explicit Berry-Esseen bound in the central limit theorem for nonlinear statistics with bounded differences. Some examples are provided as well.

Bounds in the mean central limit theorem, where the L1 distance is used to measure the discrepancy of the distribution Fn of a standardized sum of i.i.d. random variables with distributionG from the normal, is of some interest when the normal approximation is to be applied over some wide, and perhaps unspecified range of values. Esseen (1958) showed that the limiting value lim n→∞ n||Fn − Φ||1 ...

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.

For a supercritical branching process (Zn) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population Wn = Zn/E[Zn|ξ] to its limitW∞: we show a central limit theorem forW∞−Wn with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distrib...