Strong approximation of the empirical distribution function for absolutely regular sequences in ${\mathbb R}^d$

Jérôme Dedecker (Université Paris Descartes)
Emmanuel Rio (Université de Versailles)
Florence Merlevède (Université Paris-Est Marne-la-Vallée)

Abstract


We prove a strong approximation result with rates for the empirical process associated to an absolutely regular stationary sequence of random variables with values in ${\mathbb R}^d$. As soon as the absolute regular coefficients of the sequence decrease more rapidly than $n^{1-p} $ for some $p \in ]2,3]$, we show that the error of approximation between the empirical process and a two-parameter Gaussian process is of order $n^{1/p} (\log n)^{\lambda(d)}$ for some positive $\lambda(d)$ depending on $d$, both in ${\mathbb L}^1$ and almost surely. The power of $n$ being independent of the dimension, our results are even new in the independent setting, and improve earlier results. In addition, for absolutely regular sequences, we show that the rate of approximation is optimal up to the logarithmic term.

 


Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-56

Publication Date: January 14, 2014

DOI: 10.1214/EJP.v19-2658

References

  • Beck, József. Lower bounds on the approximation of the multivariate empirical process. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 2, 289--306. MR0799151
  • Bentkus, V. Yu.; Lyubinskas, K. Rate of convergence in the invariance principle in Banach spaces. (Russian) Litovsk. Mat. Sb. 27 (1987), no. 3, 423--434. MR0925348
  • Borisov, I. S. Approximation of empirical fields that are constructed from vector observations with dependent coordinates. (Russian) Sibirsk. Mat. Zh. 23 (1982), no. 5, 31--41, 222. MR0673536
  • Borisov, I. S. An approach to the problem of approximation of distributions of sums of independent random elements. (Russian) Dokl. Akad. Nauk SSSR 272 (1983), no. 2, 271--275. MR0724499
  • Borisov, I. S. An accuracy of Gaussian approximation of sum distribution of independent random variables in Banach spaces. Probability theory and mathematical statistics (Kyoto, 1986), 28--54, Lecture Notes in Math., 1299, Springer, Berlin, 1988. MR0935975
  • Castelle, Nathalie; Laurent-Bonvalot, Françoise. Strong approximations of bivariate uniform empirical processes. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 4, 425--480. MR1632841
  • Cirelʹson, B. S.; Ibragimov, I. A.; Sudakov, V. N. Norms of Gaussian sample functions. Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), pp. 20--41. Lecture Notes in Math., Vol. 550, Springer, Berlin, 1976. MR0458556
  • Csörgő, M.; Révész, P. A new method to prove strassen type laws of invariance principle. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1975), no. 4, 261--269. MR1554018
  • Csörgő, M.; Révész, P. A strong approximation of the multivariate empirical process. Studia Sci. Math. Hungar. 10 (1975), no. 3-4, 427--434 (1978). MR0515344
  • Csörgő, Miklós; Horváth, Lajos. A note on strong approximations of multivariate empirical processes. Stochastic Process. Appl. 28 (1988), no. 1, 101--109. MR0936377
  • Delyon B.: Limit theorem for mixing processes. Tech. report IRISA, Rennes 1, 546, (1990).
  • Dedecker, Jérôme; Merlevède, Florence; Rio, Emmanuel. Strong approximation results for the empirical process of stationary sequences. Ann. Probab. 41 (2013), no. 5, 3658--3696. MR3127895
  • Dedecker, Jérôme; Prieur, Clémentine; Raynaud De Fitte, Paul. Parametrized Kantorovich-Rubinštein theorem and application to the coupling of random variables. Dependence in probability and statistics, 105--121, Lecture Notes in Statist., 187, Springer, New York, 2006. MR2283252
  • Dhompongsa, Sompong. A note on the almost sure approximation of the empirical process of weakly dependent random vectors. Yokohama Math. J. 32 (1984), no. 1-2, 113--121. MR0772909
  • Doukhan, Paul; Portal, Frédéric. Principe d'invariance faible pour la fonction de répartition empirique dans un cadre multidimensionnel et mélangeant. (French) [The weak invariance principle for the empirical distribution function in the multidimensional mixing case] Probab. Math. Statist. 8 (1987), 117--132. MR0928125
  • Dudley, R. M. Central limit theorems for empirical measures. Ann. Probab. 6 (1978), no. 6, 899--929 (1979). MR0512411
  • Dudley, R. M.; Philipp, Walter. Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. Verw. Gebiete 62 (1983), no. 4, 509--552. MR0690575
  • Götze, F. On the rate of convergence in the central limit theorem in Banach spaces. Ann. Probab. 14 (1986), no. 3, 922--942. MR0841594
  • Horn, Roger A.; Johnson, Charles R. Topics in matrix analysis. Cambridge University Press, Cambridge, 1991. viii+607 pp. ISBN: 0-521-30587-X MR1091716
  • Kantorovitch, L. On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37, (1942). 199--201. MR0009619
  • Kantorovič, L. V.; Rubinšteĭn, G. Š. On a space of completely additive functions. (Russian) Vestnik Leningrad. Univ. 13 1958 no. 7 52--59. MR0102006
  • Kiefer, J. Skorohod embedding of multivariate RV's, and the sample DF. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), no. 1, 1--35. MR1554013
  • Komlós, J.; Major, P.; Tusnády, G. An approximation of partial sums of independent ${\rm RV}$'s and the sample ${\rm DF}$. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 111--131. MR0375412
  • Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9 MR1102015
  • Liu, Jun S. Siegel's formula via Stein's identities. Statist. Probab. Lett. 21 (1994), no. 3, 247--251. MR1310101
  • Massart, Pascal. Strong approximation for multivariate empirical and related processes, via KMT constructions. Ann. Probab. 17 (1989), no. 1, 266--291. MR0972785
  • Merlevède, Florence; Rio, Emmanuel. Strong approximation of partial sums under dependence conditions with application to dynamical systems. Stochastic Process. Appl. 122 (2012), no. 1, 386--417. MR2860454
  • Neumann, Michael H. A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics. ESAIM Probab. Stat. 17 (2013), 120--134. MR3021312
  • Peligrad, Magda; Utev, Sergey. Central limit theorem for stationary linear processes. Ann. Probab. 34 (2006), no. 4, 1608--1622. MR2257658
  • Philipp, Walter; Pinzur, Laurence. Almost sure approximation theorems for the multivariate empirical process. Z. Wahrsch. Verw. Gebiete 54 (1980), no. 1, 1--13. MR0595473
  • Rio, Emmanuel. Local invariance principles and their application to density estimation. Probab. Theory Related Fields 98 (1994), no. 1, 21--45. MR1254823
  • Rio, Emmanuel. Vitesses de convergence dans le principe d'invariance faible pour la fonction de répartition empirique multivariée. (French) [Rates of convergence in the weak invariance principle for the multivariate empirical distribution function] C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 2, 169--172. MR1373756
  • Rio, Emmanuel. Processus empiriques absolument réguliers et entropie universelle. (French) [Absolutely regular empirical processes and universal entropy] Probab. Theory Related Fields 111 (1998), no. 4, 585--608. MR1641838
  • Rio, Emmanuel. Théorie asymptotique des processus aléatoires faiblement dépendants. (French) [Asymptotic theory of weakly dependent random processes] Mathématiques & Applications (Berlin) [Mathematics & Applications], 31. Springer-Verlag, Berlin, 2000. x+169 pp. ISBN: 3-540-65979-X MR2117923
  • Rosenblatt, M. A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U. S. A. 42 (1956), 43--47. MR0074711
  • Volkonskiĭ, V. A.; Rozanov, Yu. A. Some limit theorems for random functions. I. Theor. Probability Appl. 4 1959 178--197. MR0121856
  • Rüschendorf, Ludger. The Wasserstein distance and approximation theorems. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 1, 117--129. MR0795791
  • Sakhanenko, A. I. Simple method of obtaining estimates in the invariance principle. Probability theory and mathematical statistics (Kyoto, 1986), 430--443, Lecture Notes in Math., 1299, Springer, Berlin, 1988. MR0936018
  • Sakhanenko, Alexander I. A new way to obtain estimates in the invariance principle. High dimensional probability, II (Seattle, WA, 1999), 223--245, Progr. Probab., 47, Birkhäuser Boston, Boston, MA, 2000. MR1857325
  • Skorohod, A. V. On a representation of random variables. (Russian) Teor. Verojatnost. i Primenen. 21 (1976), no. 3, 645--648. MR0428369
  • Tusnády, G. A remark on the approximation of the sample $DF$ in the multidimensional case. Period. Math. Hungar. 8 (1977), no. 1, 53--55. MR0443045
  • Wu, Wei Biao. Strong invariance principles for dependent random variables. Ann. Probab. 35 (2007), no. 6, 2294--2320. MR2353389
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.