Generalization of the Glivenko–Cantelli theorem to finite-dimensional distributions of ergodic homogeneous random fields

Arkady Tempelman

Statistics & Probability Letters, 2023, vol. 195, issue C

Abstract: Let X(t),t∈Zm×Rk, be an ergodic stationary random processes or an ergodic homogeneous random field (k+m≥1). We prove that the distribution function of each random vector (X(t1),…,X(ts)) can be a.s. arbitrary fine uniformly approximated by the empirical distribution functions, if, e.g., in their construction increasing bounded convex sets Tn with infinitely increasing intrinsic diameters are used. We consider also the case when X is observed on a mixing homogeneous countable random set Sm(ω)⊂Rm (e.g., on a Poisson random set). These results open a way to consistent statistical inference on finite-dimensional distribution functions of ergodic processes and fields.

Keywords: Glivenko–Cantelli theorem; Stationary random process; Homogeneous random field; Distribution function; Empirical distribution function (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spl.2022.109767

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