Nonuniform estimate of the rate of convergence in the CLT with stable limit distribution
Let \(X_ 1,X_ 2,...\in {\mathbb{R}}\) be independent random variables with the same distribution F. Let Y denote the strictly stable random variable with exponent \(0<\alpha \leq 2\) and distribution G. The authors prove that \[ \sup \{| r|^ tP(X_ 1+...+X_ n>rn^{1/\alpha})- P(Y>r);\quad r\in R\}\leq c_ o(t,Y)n^{1-t/\alpha}\int_{{\mathbb{R}}}| x|^ t| F-G| (dx), \] for \(\alpha <t\leq 1+\alpha\). The constant \(c_ 0(t,Y)\) depends only on t and Y and can be estimated by the function \[ B(i,s)=\sup_{u}(| p^{(i)}(u)| (1+| u|^ s)), \] where \(p^{(i)}\) is the ith derivative of the density of Y, \(i=1,2,..\). and \(s\geq 0\).
scientific article; zbMATH DE number 4128152
scientific article