# Limit Laws for Sums of Logarithms of k-Spacings

## TL;DR

This paper proves that the sum of logarithms of k-spacings in a sample follows a normal distribution as the sample size grows.

## Contribution

The paper establishes asymptotic normality for sums of logarithms of k-spacings under general conditions.

## Key findings

- The sum of logarithms of k-spacings is asymptotically normal.
- The result holds for a wide class of distribution functions with Riemann integrable densities.
- The findings extend and complete prior research on k-spacings.

## Abstract

Let Z=Z1,…,Zn be an i.i.d. sample from the absolutely continuous distribution function F(z):=P(Z≤z), with density f(z):=ddzF(z). Let Z1,n<…<Zn,n be the order statistics generated by Z1,…,Zn. Let Z0,n=a:=inf{z:F(z)>0} and Zn+1,n=b:=sup{z:F(z)<1} denote the end-points of the common distribution of these observations, and assume that the density f is Riemann integrable and bounded away from 0 over each interval [a′,b′]⊂(a,b). For a specified k≥1, we establish the asymptotic normality of the sum of logarithms of the k-spacings Zi+k,n−Zi−1,n for i=1,…,n−k+2. Our results complete previous investigations in the literature conducted by Blumenthal, Cressie, Shao and Hahn, and the references therein.