Non standard functional limit laws for the increments of the compound empirical distribution function

Myriam Maumy-Bertrand (1) , Davit Varron (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

Let (Y i , Z i) i≥1 be a sequence of independent, identically distributed (i.i.d.) random vectors taking values in R k × R d , for some integers k and d. Given z ∈ R d , we provide a nonstandard functional limit law for the sequence of functional increments of the compound empirical process, namely ∆n,c(hn, z, ·) := 1 nhn n i=1 1 [0,·) Z i − z hn 1/d Y i. Provided that nhn ∼ c log n as n → ∞, we obtain, under some natural conditions on the conditional exponential moments of Y | Z = z, that ∆n,c(hn, z, ·) Γ almost surely, where denotes the clustering process under the sup norm on [0, 1) d. Here, Γ is a compact set that is related to the large deviations of certain compound Poisson processes.

Mots clés: Poisson process; Large de-viations; and phrases: Empirical processes

Domaines: Statistiques [math.ST]

Myriam Maumy-Bertrand, Davit Varron. Non standard functional limit laws for the increments of the compound empirical distribution function. Electronic Journal of Statistics , 2010, 4, pp.1324-1344. ⟨10.1214/09-EJS381⟩. ⟨hal-01158049⟩