Irregular sets and Central Limit Theorems for dependent triangular arrays

2009/11/05 by Beatriz Marrón, Beatriz Marron, Marron, Beatriz +2
Decision Sciences · Mathematics · #Probability and Risk Models #stat.ME

paper · pdf · doi:10.48550/arxiv.0911.1117

14 pages

arxiv created 2009/11/05 · arxiv updated 2009/12/01

Abstract

In previous papers, we studied the asymptotic behaviour of S_N(A,X)=(2N+1)^-d/2∑_{n ∈ AN} X_n, where X is a centered, stationary and weakly dependent random field, and A_N=A ∩ [-N,N]^d, A ⊂ ℤ^d. This leads to the definition of asymptotically measurable sets, which enjoy the property that S_N(A;X) has a Gaussian weak limit for any X belonging to a certain class. Here we extend this type of results to the case of weakly dependent triangular arrays and present an application of this technique to regression models. Indeed, we prove that CLT and related results hold for X_n^N=φ(ξ_n^N,Y_n^N), n ∈ ℤ^d, where φ satisfies certain regularity conditions, ξ and Y are independent random fields, ξ is weakly dependent and Y satisfies some Strong Law of Large Numbers.

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