Non-uniform spacings processes

We provide a joint strong approximation of the uniform spacings empirical process and of the uniform quantile process by sequences of independent Gaussian processes. This allows us to obtain an explicit description of the limiting Gaussian process generated by the sample spacings from a non-uniform distribution. It is of the form B(t) + (1-σF) {(1-t)log(1-1)} ∫01, for 0 ≤ t ≤ 1, where {B(t):0 ≤ t ≤ 1} denotes a Brownian bridge, and where σF2=Var(log f(X)) is a factor depending upon the underlying distribution function F(·)=P(X ≤ x) through its density f (x)= d/dx F(x). We provide a strong approximation of the non-uniform spacings processes by replicæ of this Gaussian process, with limiting sup-norm rate OP(n-1/8(log n)1/2). The limiting process reduces to a Brownian bridge if and only if σ2F, which is the case when the sample observations are exponential. For uniform spacings, we get σ2F=0, which is in agreement with the results of Beirlant (In: Limit theorems in probability and statistics, Proc Coll Math Soc J Bolyai, vol 36, Akadémiai Kiadó, Budapest, pp 77-80, 1984), and Aly et al. (Z Wahrsch Verw Gebiete 66:461-484, 1984). © 2011 Springer Science+Business Media B.V.