Geodesic
An exponential inequality for $U$-statistics of i.i.d. data
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 508-533 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We establish an exponential inequality for degenerated $U$-statistics of order $r$ of independent and identically distributed (i.i.d.) data. This inequality gives a control of the tail of the maxima absolute values of the $U$-statistic by the sum of the two terms: an exponential term and one involving the tail of $h(X_1,\dots,X_r)$. We also give a version for not necessarily degenerated $U$-statistics having a symmetric kernel and furnish an application to the convergence rates in the Marcinkiewicz law of large numbers. Application to the invariance principle in Hölder spaces is also considered.
Keywords: $U$-statistics, exponential inequality. 
@article{TVP_2021_66_3_a5,
     author = {D. Giraudo},
     title = {An exponential inequality for $U$-statistics of i.i.d. data},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {508--533},
     year = {2021},
     volume = {66},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a5/}
}
TY  - JOUR
AU  - D. Giraudo
TI  - An exponential inequality for $U$-statistics of i.i.d. data
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2021
SP  - 508
EP  - 533
VL  - 66
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a5/
LA  - ru
ID  - TVP_2021_66_3_a5
ER  - 
%0 Journal Article
%A D. Giraudo
%T An exponential inequality for $U$-statistics of i.i.d. data
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2021
%P 508-533
%V 66
%N 3
%U http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a5/
%G ru
%F TVP_2021_66_3_a5
D. Giraudo. An exponential inequality for $U$-statistics of i.i.d. data. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 508-533. http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a5/

[1] M. A. Arcones, “A Bernstein-type inequality for U-statistics and U-processes”, Statist. Probab. Lett., 22:3 (1995), 239–247 | DOI | MR | Zbl

[2] M. A. Arcones, E. Giné, “Limit theorems for $U$-processes”, Ann. Probab., 21:3 (1993), 1494–1542 | DOI | MR | Zbl

[3] L. E. Baum, M. Katz, “Convergence rates in the law of large numbers”, Trans. Amer. Math. Soc., 120 (1965), 108–123 | DOI | MR | Zbl

[4] T. C. Christofides, “Probability inequalities with exponential bounds for $U$-statistics”, Statist. Probab. Lett., 12:3 (1991), 257–261 | DOI | MR | Zbl

[5] J. Dedecker, F. Merlevède, “A deviation bound for $\alpha$-dependent sequences with applications to intermittent maps”, Stoch. Dyn., 17:1 (2017), 1750005, 27 pp. | DOI | MR | Zbl

[6] Xiequan Fan, I. Grama, Quansheng Liu, “Large deviation exponential inequalities for supermartingales”, Electron. Commun. Probab., 17 (2012), 59, 8 pp. | DOI | MR | Zbl

[7] Xiequan Fan, I. Grama, Quansheng Liu, “Exponential inequalities for martingales with applications”, Electron. J. Probab., 20 (2015), 1, 22 pp. | DOI | MR | Zbl

[8] E. Giné, R. Latała, J. Zinn, “Exponential and moment inequalities for $U$-statistics”, High dimensional probability (Seattle, WA, 1999), v. 2, Progr. Probab., 47, Birkhäuser Boston, Boston, MA, 2000, 13–38 | DOI | MR | Zbl

[9] E. Giné, J. Zinn, “Marcinkiewicz type laws of large numbers and convergence of moments for $U$-statistics”, Probability in Banach spaces (Brunswick, ME, 1991), v. 8, Progr. Probab., 30, Birkhäuser Boston, Boston, MA, 1992, 273–291 | DOI | MR | Zbl

[10] D. Giraudo, “Holderian weak invariance principle under a Hannan type condition”, Stochastic Process. Appl., 126:1 (2016), 290–311 | DOI | MR | Zbl

[11] D. Giraudo, “Holderian weak invariance principle for stationary mixing sequences”, J. Theoret. Probab., 30:1 (2017), 196–211 | DOI | MR | Zbl

[12] W. Hoeffding, “Probability inequalities for sums of bounded random variables”, J. Amer. Statist. Assoc., 58:301 (1963), 13–30 | DOI | MR | Zbl

[13] Ch. Houdré, P. Reynaud-Bouret, “Exponential inequalities, with constants, for $U$-statistics of order two”, Stochastic inequalities and applications, Progr. Probab., 56, Birkhäuser, Basel, 2003, 55–69 | DOI | MR | Zbl

[14] P. N. Kokic, “Rates of convergence in the strong law of large numbers for degenerate $U$-statistics”, Statist. Probab. Lett., 5:5 (1987), 371–374 | DOI | MR | Zbl

[15] E. Lesigne, D. Volný, “Large deviations for martingales”, Stochastic Process. Appl., 96:1 (2001), 143–159 | DOI | MR | Zbl

[16] Kuang Hsien Lin, “Convergence rate and the first exit time for $U$-statistics”, Bull. Inst. Math. Acad. Sinica, 9:1 (1981), 129–143 | MR | Zbl

[17] P. Major, “On a multivariate version of Bernstein's inequality”, Electron. J. Probab., 12 (2007), 966–988 | DOI | MR | Zbl

[18] P. Major, “Estimation of multiple random integrals and $U$-statistics”, Mosc. Math. J., 10:4 (2010), 747–763 | DOI | MR | Zbl

[19] A. Mandelbaum, M. S. Taqqu, “Invariance principle for symmetric statistics”, Ann. Statist., 12:2 (1984), 483–496 | DOI | MR | Zbl

[20] F. Merlevède, M. Peligrad, S. Utev, “Recent advances in invariance principles for stationary sequences”, Probab. Surv., 3 (2006), 1–36 | DOI | MR | Zbl

[21] A. Račkauskas, Ch. Suquet, “Necessary and sufficient condition for the Lamperti invariance principle”, Teoriya Imovir. ta Matem. Statist., 68 (2003), 115–124 ; Theory Probab. Math. Statist., 68 (2004), 127–137 | MR | Zbl

[22] A. Račkauskas, Ch. Suquet, “Necessary and sufficient condition for the functional central limit theorem in Hölder spaces”, J. Theoret. Probab., 17:1 (2004), 221–243 | DOI | MR | Zbl

[23] L. Rüschendorf, “Ordering of distributions and rearrangement of functions”, Ann. Probab., 9:2 (1981), 276–283 | DOI | MR | Zbl

[24] Ch. Suquet, “Tightness in Schauder decomposable Banach spaces”, Proceedings of the St. Petersburg Mathematical Society, v. 5, Amer. Math. Soc. Transl. Ser. 2, 193, Amer. Math. Soc., Providence, RI, 1999, 201–224 | DOI | MR | Zbl

[25] H. Teicher, “On the Marcinkiewicz–Zygmund strong law for $U$-statistics”, J. Theoret. Probab., 11:1 (1998), 279–288 | DOI | MR | Zbl

[26] Qiying Wang, “Bernstein type inequalities for degenerate $U$-statistics with applications”, Chinese Ann. Math. Ser. B, 19:2 (1998), 157–166 | MR | Zbl