Asymptotic expansions in the integral and local limit theorems in Banach spaces with applications to \(\omega\)-statistics
Two different but closely related topics are considered. The first one is the asymptotic analysis in the central limit theorem in Banach spaces including estimates of the convergence rate and asymptotic expansions of the remainder terms. The second topic deals with asymptotic analysis of \(\omega\)-statistics, \[ \omega^ p_ n(q) = n^{p/2} \int^ 1_ 0 [F_ n(x) - x]^ p q(x)dx, \] where \(p \geq 2\) is an even number, \(q : [0,1] \to [0,\infty]\) is a measurable weight function, and \(F_ n(x) = n^{-1} \sum^ n_{k = 1} I\{U_ i \leq x\}\) is the empirical distribution function based on a sample \(U_ 1,\dots,U_ n\) taken from the uniform on (0,1) distribution. The main purpose of the paper is to link these two topics. Results on the asymptotic analysis in the CLT proved in abstract Banach spaces are applied to statistics. This is designed so that the conditions for a particular asymptotic behavior of the distribution of statistic \(\omega^ p_ n(q)\) are expressed in terms of the weight function \(q\) and are easy to verify. On the other hand a particular investigation of statistics like \(\omega\)-statistics leads to some new investigations in the CLT theory in abstract Banach spaces. Several new results are proved on the asymptotic expansions in the CLT in Banach spaces for the distribution of a polynomial of sum. The density and higher order derivatives are investigated, too. The paper gives a good account on both topics as well as detailed references.
scientific article; zbMATH DE number 563113
scientific article