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.

In this paper we study the estimation of a quantile function based on left truncated and right censored data by the kernel smoothing method. Asymptotic normality and a Berry-Esseen type bound for the kernel quantile estimator are derived. Monte Carlo studies are conducted to compare the proposed estimator with the PL-quantile estimator.

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.

Berry Esseen type bounds to the normal, based on zeroand size-bias couplings, are derived using Stein’s method. The zero biasing bounds are illustrated with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds ...

For sum of independent and identically distributed (i.i.d.) random variables {Xi}i=1, the Berry-Esseen theorem states that sup y∈< ∣∣∣∣Pr { 1 sn (X1 + X2 + · · ·+ Xn) ≤ y } − Φ(y) ∣∣∣∣ ≤ C ρ σ3 √ n , where σ and ρ are respectively the variance and the absolute third moment of the parent distribution, Φ(·) is the unit normal cumulative distribution function, and C is an absolute constant. In thi...

In this paper, we establish optimal Berry–Esseen bounds for the generalized U-statistics. The proof is based on a new theorem exchangeable pair approach by Stein’s method under general linearity condition setting. As applications, an convergence rate of normal approximation subgraph counts in Erdos?–Rényi graphs and graphon-random graph obtained.