Speed of convergence of classical empirical processes in \(p\)-variation norm. (English) Zbl 1010.62039

Summary: Let \(F\) be any distribution function on \(\mathbb{R}\), and \(F_n\) be the \(n\) th empirical distribution function based on variables 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 \(\|\cdot \|_{[p]}\), \[ E\bigl\|n^{1/2} (F_n-F)-B_n\circ F\bigr\|_{[p]}\leq C(p) n^{(2-p)/(2p)}, \] where \((B_n\circ F)(x)= B_n(F(x))\). The expectation can be replaced by an Orlicz norm of exponential order. Conversely, if \(F\) is continuous, then for any stochastie process \(V(t,\omega)\) continuous in \(t\) for almost all \(\omega\), such as \(B_n\circ F\), summation over \(n\) distinct jumps shows that \[ \bigl\|n^{1/2}(F_n-F) -V\bigr\|_{[p]} \geq n^{(2-p)/(2p)}, \] 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, [J. Komlós et al., Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029)], as for the supremum norm \((p= \infty)\).


62G30 Order statistics; empirical distribution functions
60F17 Functional limit theorems; invariance principles


Zbl 0308.60029
Full Text: DOI


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