An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes

Marcin Dudziński

Abstract


Let: \(\mathbf{Y=}\left( \mathbf{Y}_{i}\right)\), where \(\mathbf{Y}_{i}=\left( Y_{i,1},...,Y_{i,d}\right)\), \(i=1,2,\dots \), be a \(d\)-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf \(F\), and \(F_{n}\left( \mathbf{x}\right) :=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}\left(Y_{i,1}\leq x_{1},\dots ,Y_{i,d}\leq x_{d}\right)\) denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process \(B_{n}=\sqrt{n}\left( F_{n}-F\right)\) under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

Keywords


Almost sure central limit theorem; weak dependence; empirical processes; copulas

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References


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DOI: http://dx.doi.org/10.17951/a.2017.71.1.11
Date of publication: 2017-06-30 17:33:53
Date of submission: 2017-06-30 11:59:09


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