Convergence rates in the law of large numbers for Banach-valued dependent variables

J. Dedecker   ; F. Merlevede

Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 3, pp. 562-587

Résumé

We extend Marcinkievicz–Zygmund strong laws of large numbers for martingales to weakly dependent random variables with values in smooth Banach spaces. The conditions are expressed in terms of conditional expectations. In the case of Hilbert spaces, we show that our conditions are weaker than optimal ones for strongly mixing sequences (which were previously known for real-valued variables only). As a consequence, we give rates of convergence for Cramér–von Mises statistics and for the empirical estimator of the covariance operator of a Hilbert-valued autoregressive process.

Détail

Keywords: smooth Banach spaces, Hilbert spaces, Marcinkievicz–Zygmund strong laws of large numbers, almost sure convergence, weak dependence
Mots-clés : martingales, Cramér–von Mises statistics.

Bibliographie

[1] de Acosta A., “Inequalities for B-valued random vectors with applications to the strong law of large numbers”, Ann. Probab., 9:1 (1981), 157–161 | DOI | MR | Zbl

[2] Assouad P., “Espaces $p$-lisses et $q$-convexes, inégalités de Burkholder”, Séminaire Maurey–Schwartz 1974–1975: Espaces $L^p$, applications radonifiantes et géométrie des espaces de Banach, Centre Math., École Polytech., Paris, 1975, Exp. no XV, 8 pp. | MR

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

[4] Berbee H. C. P., “Convergence rates in the strong law for bounded mixing sequences”, Probab. Theory Related Fields, 74:2 (1987), 255–270 | DOI | MR | Zbl

[5] Bosq D., Linear Processes in Function Spaces. Theory and Applications, Lecture Notes in Statist., 149, Springer-Verlag, New York, 2000, 283 pp. | MR | Zbl

[6] Bradley R. C., Introduction to strong mixing conditions, Technical Report. V. 1, Dept. of Math., Indiana Univ., Custom Publishing of I. U., Bloomington, 2002 | MR

[7] Dedecker J., Doukhan P., “A new covariance inequality and applications”, Stochastic Process. Appl., 106:1 (2003), 63–80 | DOI | MR | Zbl

[8] Dedecker J., Merlevède F., “The conditional central limit theorem in Hilbert spaces”, Stochastic Process. Appl., 108:2 (2003), 229–262 | MR | Zbl

[9] Dedecker J., Prieur C., “New dependence coefficients. Examples and applications to statistics”, Probab. Theory Related Fields, 132:2 (2005), 203–236 | DOI | MR | Zbl

[10] Elton J., “A law of large numbers for identically distributed martingale differences”, Ann. Probab., 9:3 (1981), 405–412 | DOI | MR | Zbl

[11] Gordin M. I., “O tsentralnoi predelnoi teoreme dlya statsionarnykh protsessov”, Dokl. AN SSSR, 188:4 (1969), 739–741 | MR | Zbl

[12] Hall P., Heyde C. C., Martingale Limit Theory and its Applications, Academic Press, New York–London, 1980, 308 pp. | MR

[13] Hipp C., “Convergence rates of the strong law for stationary mixing sequences”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 49:1 (1979), 49–62 | DOI | MR | Zbl

[14] Lai T. L., “Convergence rates and $r$-quick versions of the strong law for stationary mixing sequences”, Ann. Probab., 5 (1977), 693–706 | DOI | MR | Zbl

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

[16] McLeish D. L., “A maximal inequality and dependent strong laws”, Ann. Probab., 3:5 (1975), 829–839 | DOI | MR | Zbl

[17] Merlevède F., “On the central limit theorem and its weak invariance principle for strongly mixing sequences with values in a Hilbert space via martingale approximation”, J. Theoret. Probab., 16:3 (2003), 625–653 | DOI | MR | Zbl

[18] Mourier E., “Éléments aléatoires dans un espace de Banach”, Ann. Inst. H. Poincaré, 13 (1953), 161–244 | MR | Zbl

[19] Peligrad M., “Convergence rates of the strong law for stationary mixing sequences”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 70:2 (1985), 307–314 | DOI | MR | Zbl

[20] Peligrad M., “The $r$-quick version of the strong law for stationary $\varphi$-mixing sequences”, Almost Everywhere Convergence (Columbus, 1988), Academic Press, Boston, 1989, 335–348 | MR

[21] Pisier G., “Martingales with values in uniformly convex spaces”, Israel J. Math., 20:3–4 (1975), 326–350 | DOI | MR | Zbl

[22] Rio E., “A maximal inequality and dependent Marcinkiewicz–Zygmund strong laws”, Ann. Probab., 23:2 (2002), 918–937 | DOI | MR

[23] Rio E., Théorie asymptotique des processus aléatoires faiblement dépendants, Math. Appl. (Berlin), 31, Springer-Verlag, Berlin, 2000, 169 pp. | MR | Zbl

[24] Rosenblatt M., “A central limit theorem and a strong mixing condition”, Proc. Natl. Acad. Sci. USA, 42 (1956), 43–47 | DOI | MR | Zbl

[25] Rudin W., Real and Complex Analysis, McGraw-Hill, New York, 1987, 416 pp. | MR | Zbl

[26] Shao Q.-M., “Complete convergence for $\alpha$-mixing sequences”, Statist. Probab. Lett., 16:4 (1993), 279–287 | DOI | MR | Zbl

[27] Woyczyński W. A., “A central limit theorem for martingales in Banach spaces”, Bull. Acad. Polon. Sci., 23:8 (1975), 917–920 | MR | Zbl

[28] Woyczyński W. A., “Laws of large numbers for vector-valued martingales”, Bull. Acad. Polon. Sci., 23:11 (1975), 1199–1201 | MR

[29] Woyczyński W. A., “Asymptotic behavior of martingales in Banach spaces. II”, Martingale Theory in Harmonic Analysis and Banach Spaces (Cleveland, 1981), Lecture Notes in Math., 939, Springer, Berlin–New York, 1982, 216–225 | MR