Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space

We prove the asymptotic independence of the empirical process $α_n = \sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $β_n = n (F_n(τ+\frac{\cdot}{n})-F_n(τ))$, where $F$ is an arbitrary cdf, differentiable at some point $τ$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n \in N$, there is a deterministic correspondence between $α_n$ and $β_n$. Precisely, we show that the pair $(α_n,β_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge and a two-sided Poisson process. Since these processes have jumps, in particular if $F$ itself has jumps, the Skorokhod product space $D(R) \times D(R)$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D(R) \times D(R)$ by establishing the classical principle, devised by Yu. V. Prokhorov, that finite-dimensional convergence and tightness imply weak convergence. Several tightness criteria are given. Finally, the convergence of the pair $(α_n,β_n)$ implies convergence of each of its components, thus, in passing, we provide a thorough proof of these known convergence results in a very general setting. In fact, the condition on $F$ to be differentiable in at least one point is only required for $β_n$ to converge and can be further weakened.

No reviews yet.

No comments yet.