Non-uniform spacings processes

Abstract

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 \({B(t)+(1-\sigma_F) \{(1-t)\log(1-t)\} \int_{0}^1\{B(s)/(1-s)\}ds}\), for 0 ≤ t ≤ 1, where {B(t):0 ≤ t ≤ 1} denotes a Brownian bridge, and where \({\sigma^2_F={\rm Var}({\rm log}\,f(X))}\) is a factor depending upon the underlying distribution function \({F(\cdot)=\mathbb{P}(X\leq x)}\) through its density \({f(x)=\frac{d}{dx}F(x)}\). We provide a strong approximation of the non-uniform spacings processes by replicae of this Gaussian process, with limiting sup-norm rate \({O_{\mathbb{P}}(n^{-1/8}(\log n)^{1/2})}\). The limiting process reduces to a Brownian bridge if and only if \({\sigma^2_F=1}\), which is the case when the sample observations are exponential. For uniform spacings, we get \({\sigma^{2}_F=0}\), which is in agreement with the results of Beirlant (In: Limit theorems in probability and statistics, Proc Coll Math Soc J Bolyai, vol 36, Akademiai Kiado, Budapest, pp 77–80, 1984), and Aly et al. (Z Wahrsch Verw Gebiete 66:461–484, 1984).