A general form of the law of the iterated logarithm for the weighted multivariate empirical process
Consider a sequence of i.i.d. random vectors \(X_ 1,X_ 2,..\). which are uniformly distributed on \([0,1]^ d\), \(d\geq 1\). For \(t=(t_ 1,...,t_ d)\), let \(| t| =\prod^{d}_{i=1}t_ i\). For \(c>0\) and \(n\geq 1\) let \(a_ n=c \log \log n/n\) and let \(F_ n(t)\) be the empirical distribution function of \(X_ 1,...,X_ n\). The main result establishes the almost sure value of \[ \limsup_{n\to \infty}\sup_{a_ n\leq | t| \leq 1}n^{\nu}| F_ n(t)- | t| | /((\log \log n)^{\nu}| t|^{1-\nu}) \] where \(0\leq \nu \leq 1/2\). The author indicates that this theorem contains all known results of this type.
scientific article; zbMATH DE number 3986290
scientific article