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

2012/01/26 by Davit Varron, Varron, Davit, Myriam Maumy‐Bertrand +2
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1201.5528

arxiv created 2012/01/26 · openalex publication_date 2012/01/26 · arxiv updated 2012/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (Y_i,Z_i)_{i≥ 1} be a sequence of independent, identically distributed (i.i.d.) random vectors taking values in \RRR^k×\RRR^d, for some integers k and d. Given z∈ \RRR^d, we provide a nonstandard functional limit law for the sequence of functional increments of the compound empirical process, namely \mathbfΔ_n,\cc(h_n,z,⋅):= (1)/(nh_n)\sliin 1_[0,⋅)\poo \fracZ_i-zh_n^1/d\pff Y_i. Provided that nh_n∼ clog n as \nif, we obtain, under some natural conditions on the conditional exponential moments of Y| Z=z, that \mathbfΔ_n,\cc(h_n,z,⋅)\leadsto \Gamalmost surely, where \leadsto denotes the clustering process under the sup norm on \Idd. Here, \Gam is a compact set that is related to the large deviations of certain compound Poisson processes.

Citations

• A nonstandard uniform functional limit law for the increments of the multivariate empirical distribution function

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