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We prove the asymptotic independence of the empirical process $\alpha_n =\sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $\beta_n =n (F_n(\tau+\frac{\cdot}{n})-F_n(\tau))$, where $F$ is an arbitrary cdf,differentiable at some point $\tau$, and $F_n$ the corresponding empricial cdf.This seems rather counterintuitive, since, for every $n \in N$, there is adeterministic correspondence between $\alpha_n$ and $\beta_n$. Precisely, weshow that the pair $(\alpha_n,\beta_n)$ converges in law to a limit havingindependent components, namely a time-transformed Brownian bridge and atwo-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 theadequate choice for modeling this convergence in. We develop a shortconvergence theory for $D(R) \times D(R)$ by establishing the classicalprinciple, devised by Yu. V. Prokhorov, that finite-dimensional convergence andtightness imply weak convergence. Several tightness criteria are given.Finally, the convergence of the pair $(\alpha_n,\beta_n)$ implies convergenceof each of its components, thus, in passing, we provide a thorough proof ofthese known convergence results in a very general setting. In fact, thecondition on $F$ to be differentiable in at least one point is only requiredfor $\beta_n$ to converge and can be further weakened.

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