Strong limits related to the oscillation modulus of the empirical process based on the k-spacing process

TL;DR

This paper extends strong limit theorems for the oscillation moduli of the empirical process to the case based on non-overlapping k-spacings from uniform iid variables, including unbounded k, and derives weak limits.

Contribution

It demonstrates that existing strong limit results for the empirical process also hold for the k-spacing based process, even when k is unbounded, with new weak limit findings.

Findings

Strong limit theorems apply to k-spacing empirical processes.

Weak limits are established for the process with unbounded k.

Results are derived from properties of gamma function increments.

Abstract

Recently, several strong limit theorems for the oscillation moduli of the empirical process have been given in the iid-case. We show that, with very slight differences, those strong results are also obtained for some representation of the reduced empirical process based on the (non-overlapping) k-spacings generated by a sequence of independent random variables (rv's) uniformly distributed on $(0,1)$. This yields weak limits for the mentioned process. Our study includes the case where the step k is unbounded. The results are mainly derived from several properties concerning the increments of gamma functions with parameters k and one.