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

Let: \(Y=( Yᵢ)\), where \(Yᵢ=( Yi,1,...,Yi,d)\), \(i=1,2, \), be a \(d\)-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf \(F\), and \(Fₙ( x) :=1/n∑ᵢ₌₁ⁿI(Yi,1≤ x₁, ,Yi,d≤ xd)\) denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process \(Bₙ=√n( Fₙ-F)\) under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.