The Annals of Probability

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

R.M. Dudley and Yen-Chen Huang

Full-text: Open access

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 $(p = \infty)$.

Article information

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

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1015345765

Digital Object Identifier
doi:10.1214/aop/1015345765

Mathematical Reviews number (MathSciNet)
MR1880235

Zentralblatt MATH identifier
1010.62039

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

Keywords
Brownian bridge Orlicz norms

Citation

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. https://projecteuclid.org/euclid.aop/1015345765


Export citation

References

  • [1] Berkes, I. and Philipp, W. (1977). An almost sure invariance principle for the empirical distribution function of mixing random variables.Wahrsch. Verw. Gebiete 41 115- 137.
  • [2] Bretagnolle, J. and Massart, P. (1989). Hungarian constructions from the nonasymptotic viewpoint. Ann. Probab. 17 239-256.
  • [3] Cs ¨org o, M. and Horv´ath, L. (1993). Weighted Approximations in Probability and Statistics. Wiley, Chichester.
  • [4] Cs ¨org o, M. and R´ev´esz, P. (1981). Strong Approximations in Probability and Statistics. Academic, New York.
  • [5] Donsker, M. D. (1952). Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23 277-281.
  • [6] Dudley, R. M. (1992). Fr´echet differentiability, p-variation and uniform Donsker classes. Ann. Probab. 20 1968-1982.
  • [7] Dudley, R. M. (1994). The order of the remainder in derivatives of composition and inverse operators for p-variation norms. Ann. Statist. 22 1-20.
  • [8] Dudley, R. M. (1999). An exposition of Bretagnolle and Massart's proof of the KMT theorem for the uniform empirical process. Preprint.
  • [9] Dudley, R. M. and Norvai sa, R. (1999). Differentiability of Six Operators on Nonsmooth Functions and p-Variation. Lecture Notes in Math. 1703. Springer, New York.
  • [10] Fernique, X. (1970). Int´egrabilit´e des vecteurs gaussiens. C. R. Acad. Sci. Paris S´er. A 270 1698-1699.
  • [11] Figiel, T., Hitczenko, P., Johnson, W. B., Schechtman, G. and Zinn, J. (1997). Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities. Trans. Amer. Math. Soc. 349 997-1027.
  • [12] Huang, Y.-C. (1994). Empirical distribution function statistics, speed of convergence, and p-variation. Ph. D. dissertation, MIT.
  • [13] Koml ´os, J., Major, P. and Tusn´ady, G. (1975). An approximation of partial sums of independent RV'-s, and the sample DF. I.Wahrsch. Verw. Gebiete 32 111-131.
  • [14] L´evy, P. (1937). Th´eorie de l'addition des variables al´eatoires. Gauthier-Villars, Paris.
  • [15] Luxemburg, W. A. J. and Zaanen, A. C. (1956). Conjugate spaces of Orlicz spaces. Indag. Math. 18 Akad. Wetensch. Amsterdam Proc. Ser. A 59. 217-228.
  • [16] Major, P. (1978). On the invariance principle for sums of independent identically distributed random variables. J. Multivariate Anal. 8 487-517.
  • [17] Mason, D. M. (1998). Notes on the KMT Brownian bridge approximation to the uniform empirical process. Preprint.
  • [18] Mason, D. M. and van Zwet, W. (1987). A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15 871-884.
  • [19] Paley, R. E. A. C., Wiener, N. and Zygmund, A. (1933). Notes on random functions. Math.37 647-668. [Reprinted (1976) Norbert Wiener: Collected Works (P. Masani, ed.) 1 526-557. MIT Press.
  • [20] Pinelis, I. (1994). Extremal probabilistic problems and Hotelling's T2 test under a symmetry condition. Ann. Statist. 22 357-368.
  • [21] Taylor, S. J. (1972). Exact asymptotic estimates of Brownian path variation. Duke Math. J. 39 219-241.
  • [22] Vorob'ev, N. N. (1962). Consistent families of measures and their extensions. Theory Probab. Appl. 7 147-163.
  • [23] Whittle, P. (1960). Bounds for the moments of linear and quadratic forms in independent variables. Theory Probab. Appl. 5 302-305.