Paul Deheuvels, Paris VI 


Friday, 04 May 2012, 14:30  15:30 



Paul Deheuvels, Université Paris VI Homepage NonUniform 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 nonuniform distribution. It is of the form B(t)+(1−σ_F ){(1− 4t) 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 σ_F^2= Var(log f (X)) is a factor depending upon the underlying distribution function F(·) = P(X ≤ x) through its density f (x) = d/dxF(x). We provide a strong approximation of the nonuniform spacings processes by replicæ of this Gaussian process, with limiting supnorm rate OP(n^{−1/8}(log n)^{1/2}). The limiting process reduces to a Brownian bridge if and only if σ_F^2= 1, which is the case when the sample observations are exponential. For uniform spacings, we get σ_F^2= 0, which is in agreement with the results of Beirlant (In: Limit theorems in probability and statistics, Proc Coll Math Soc J Bolyai, vol 13 36, Akadémiai Kiadó, Budapest, pp 77–80, 1984), and Aly et al. (Z Wahrsch Verw Gebiete 14 66:461–484, 1984). 
Location: plaine, building NO, 9th floor, salle des professeurs
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