Joint approximation of processes based on spacings and order statistics (Q1343585)

The author considered the joint behaviour of two empirical processes created by a sequence of i.i.d. r.v. \(\omega_ 1, \omega_ 2, \dots\). The first one is \(n^{-1} \sum_{i=1}^ n I_{S_ i/ S_{n+1}\leq x}\), and the second corresponds to the spacings empirical distribution function \(n^{-1} \sum^ n_{i=1} I_{F( n\omega_ i/ S_{n+1})\leq x}\), where \(S_ i= \omega_ 1+ \dots+ \omega_ i\). Under some conditions on the distribution of the \(\omega_ i\) the joint approximation is given. The proof is based on the approximation of the uniform empirical process by a Kiefer process and consists of two steps - - approximation of the process related to the renewal empirical distribution function and the one related to the inverse of it.