Structural improvements of nonuniform convergence rate estimates in the central limit theorem with applications to Poisson random sums
Let \(X_i\), \(i\geq1\), be a sequence of independent and identically distributed random variables with \(E(X_1)=0\), \(E(X_1^2)=1\) and \(\beta_{2+\delta}=E(|X_1|^{2+\delta})<\infty\) for some \(\delta\in(0,1]\). For every integer \(n\geq1\) and real number \(x\), set \(F_n(x)=P(X_1+\cdots+X_n<x)\) and \(\Delta_n(x)=|F_n(x\sqrt{n})-\Phi(x)|\), where \(\Phi\) denotes the distribution function of the standard normal distribution. An algorithm is presented which for each \(\delta\in(0,1]\) allows to compute an upper bound for the absolute constant \(C(\delta)\) in the inequality \[ \sup_x(1+|x|^{2+\delta})\Delta_n(x)\leq C(\delta)(\beta_{2+\delta}+1)n^{-\delta/2}\,. \] Moreover, the method permits to compute a function \(C(K,\delta)\) such that for all \(\delta\in(0,1]\), \(K\geq0\) and \(n\geq1\) \[ \sup_{|x|\geq K}|x|^{2+\delta}\Delta_n(x)\leq C(K,\delta)(\beta_{2+\delta}+1)n^{-\delta/2}\,, \] where \(C(K,\delta)\) does not depend on the common distribution of the \(X_i\)'s
scientific article; zbMATH DE number 5995565