Empirical process sampled along a stationary process

2023/01/27 by Guy Cohen, Cohen, Guy, Jean-Pierre Conze +1
Economics, Econometrics and Finance · Mathematics · #22D40 #28D05 #37A25 #37A30 #60G50 #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary: 60F05 #Probability (math.PR) #Secondary: 47B15 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2301.11576

openalex publication_date 2023/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (X_\underlineℓ)_\underlineℓ ∈ \mathbb Z^d be a real random field (r.f.) indexed by \mathbb Z^d with common probability distribution function F. Let (z_k)_k=0^∞ be a sequence in \mathbb Z^d. The empirical process obtained by sampling the random field along (z_k) is ∑_k=0^n-1 [\bf 1_X_zk ≤ s- F(s)]. We give conditions on (z_k) implying the Glivenko-Cantelli theorem for the empirical process sampled along (z_k) in different cases (independent, associated or weakly correlated random variables). We consider also the functional central limit theorem when the X_\underlineℓ's are i.i.d. These conditions are examined when (z_k) is provided by an auxiliary stationary process in the framework of ``random ergodic theorems''.

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