On the Rate of Convergence and Berry-esseen Type Theorems for a Multivariate Free Central Limit Theorem
نویسندگان
چکیده
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.
منابع مشابه
A Berry-Esseen Type Bound for the Kernel Density Estimator of Length-Biased Data
Length-biased data are widely seen in applications. They are mostly applicable in epidemiological studies or survival analysis in medical researches. Here we aim to propose a Berry-Esseen type bound for the kernel density estimator of this kind of data.The rate of normal convergence in the proposed Berry-Esseen type theorem is shown to be O(n^(-1/6) ) modulo logarithmic term as n tends to infin...
متن کاملCentral limit theorems for a supercritical branching process in a random environment
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...
متن کاملMoment bounds and central limit theorems for Gaussian subordinated arrays
A general moment bound for sums of products of Gaussian vector’s functions extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general central limit theorem for triangular arrays of nonlinear functionals of multidimensional non-stationary Gaussian sequences is proved. This theorem extends the previous results of Breuer and Major (1981), Arcones (1994) and others. A Berry-Ess...
متن کاملStein’s Method for Dependent Random Variables Occurring in Statistical Mechanics
We obtain rates of convergence in limit theorems of partial sums Sn for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the Curie-Weiss models. Under appropriate assumptions there exists a real number α, a positive real number μ, and a positive integer k such that (Sn − nα)/n converges wea...
متن کاملAsymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part
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...
متن کامل