On the rate of convergence to a stable limit law. II
[For part I see: Limit theorems in probability and statistics, 2nd Colloq., Veszprém/Hung. 1982, Vol. II, Colloq. Math. Soc. János Bolyai 36, 761-775 (1984; Zbl 0571.60031).] A theorem on the rate of convergence to a stable limit law F(x;\(\alpha\),\(\beta)\) is obtained. Let \(X_ 1,X_ 2,..\). be i.i.d. symmetric random variables with common distribution function G(x), and \[ F(x;\alpha,0)-G(x)=\phi (x)x^{-s}, \] 0\(<\alpha <1\), where \(\phi\) is, for \(x\geq x_ 0\), a bounded, continuous function with alternating sign. The author gets an estimate on the rate of convergence of the distribution functions of properly standardized partial sums of \(\{X_ n\}\) to F(x;\(\alpha\),0) under some additional assumptions. This estimate depends on \(\alpha\) and s. Some examples, given by the author, show this estimate to be stronger than related results of other authors. The author also asserts that a similar theorem in the case either \(\beta\neq 0\) or \(| \beta | \neq 1\) can be proved.
scientific article; zbMATH DE number 4050637