Weak convergence of stationary empirical processes

Weak Convergence of Stationary Empirical Processes

We offer an umbrella type result which extends the convergence of classical empirical process on the line to more general processes indexed by functions of bounded variation. This extension is not contingent on the type of dependence of the underlying sequence of random variables. As a consequence we establish the weak convergence for stationary empirical processes indexed by general classes of...

@bullet , @bullet Weak Convergence of Multidimensional Empirical Processes for Stationary ¢-mixing Processes

Empirical Processes: General Weak Convergence Theory

The lack of measurability of the empirical process with respect to the sigma-field generated by the ‘natural’ l∞ metric, as illustrated in the previous notes, needs an extension of the standard weak convergence theory that can handle situations where the converging stochastic processes may no longer be measurable (though the limit will be a tight Borel measurable random element). Of course, an ...

Weak Convergence of Blockwise Bootstrapped Empirical Processes for Stationary Random Fields with Statistical Applications

In this article, we consider a stationary α-mixing random field in IR. Under a large-sample scheme that is a mixture of the so-called “infill” and “increasing domain” asymptotics, we establish a functional central limit theorem for the empirical processes of this random field. Further, we apply a blockwise bootstrap to the samples. Under the condition that the side length of the block λl = O(λn...

An elementary proof of the weak convergence of empirical processes

This paper develops a simple technique for proving the weak convergence of a stochastic process Z̄n(g) := ∫ g dZn, indexed by functions g in some class G. The main novelty is a decoupling argument that allows to derive asymptotic equicontinuity of the process {Z̄n(g), g ∈ G} from that of the basic process {Zn(t), t ∈ R}, with Zn(t) = Z̄n(ft) and ft(x) = 1(−∞,t](x). The method leads to novel result...