The Annals of Probability

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

R.M. Dudley and Yen-Chen Huang

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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 $(p = \infty)$.

Article information

Ann. Probab., Volume 29, Number 4 (2001), 1625-1636.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 62G30: Order statistics; empirical distribution functions 60F17

Brownian bridge Orlicz norms


Huang, Yen-Chen; Dudley, R.M. Speed of Convergence of Classical Empirical Processes in $p$-variation Norm. Ann. Probab. 29 (2001), no. 4, 1625--1636. doi:10.1214/aop/1015345765.

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