Strassen's LIL for the Lorenz curve
The authors prove Strassen's law of the iterated logarithm for the Lorenz process assuming that the underlying d.f. \(F\) and its inverse \(F^{-1}\) are continuous and the moment \(EX^{2+u}\) is finite for some \(u>0\). Previously, the first author, \textit{S. Csörgö} and \textit{L. Horváth} [``An asymptotic theory for empirical reliability and concentration processes'' (1986; Zbl 0605.62105)] and \textit{R. C. Rao} and \textit{L. C. Zhao} [J. Multivariate Anal. 54, No. 2, 239-252 (1995)] proved strong invariance principle for Lorenz process using stronger assumptions on \(f\) and \(F\). Using a limit theorem for the general integrated empirical difference process and techniques of \textit{W. Vervaat} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 23, 245-253 (1972; Zbl 0238.60018)] and \textit{Polonik} (Shorack and Wellner, 1986), the authors are able to obtain sharper results for Lorenz process.
Strassen's law of the iterated logarithm
scientific article; zbMATH DE number 954537