On the weak limit law of the maximal uniform k-spacing

TL;DR

This paper provides a straightforward proof of the limit law for the maximum length of intervals between fixed numbers of uniformly distributed points, utilizing Watson's theorem on maxima of m-dependent sequences.

Contribution

It introduces a simple proof for the weak limit law of the largest uniform k-spacing, connecting it with classical results on dependent stochastic sequences.

Findings

Established the limit distribution for the maximal uniform k-spacing.

Connected the problem to Watson's theorem on maxima of m-dependent sequences.

Simplified the proof approach for the limit law of the largest spacing.

Abstract

This paper gives a simple proof of a limit theorem for the lenght of the largest interval straddling a fixed number of i.i.d. points uniformly disributed on a unit interval. The key step in our argument is a classical theorem of Watson (1954) on the maxima of m-dependent stationary stochastic sequences.