## 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 $$(p= \infty)$$.

### MSC:

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

### Keywords:

Brownian bridge; Orlicz norms

Zbl 0308.60029
Full Text:

### References:

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