Limit Laws for Sums of Logarithms of k-Spacings

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<…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.