Speed of Convergence of Classical Empirical Processes in $p$-variation Norm

Abstract

Let $F$ be any distribution function on $\mathbb{R}$, and $F_n$ be the$n$th empirical distribution function based on fariables i.i.d. ($F$). It is shown that for $2<p<\infty$ and a constant $C(p) < \infty$, not depending on $F$, on some probability space there exist $F_n$ and Brownian bridges $B_n$ such that for the Wiener-Young $p$-variation norm $\|\cdot\|_{[p]}, E\|n^{1/2}(F_n - F) - B_n \circ F\|_{[p]} \leq C(p)n^{(2-p)/(2p)}$, where $(B_n \circ F)(x))$. The expectation can be replaced by an Orlicz norm of exponential order. Conversely, if $F$ is continuous, then for any stochastic process $V(t,\omega)$ continuous in $t$ for almost all $\omega$, such as $B_n \circ F$, summation over $n$ distinct jumps shows that $\|n^{1/2}(F_n - F) - V\|_{[p]} \geq n^{(2-p)/(2p)}$, so the upper bound in expectation is best possible up to the constant $C(p)$. In the proof, $B_n$ is linked to $F_n$ by the Komlós, Major and Tusnády construction, as for the supremum norm $(p = \infty)$.