On the bounded laws of iterated logarithm in Banach space
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 19-37.

In the present paper, by using the inequality due to Talagrand's isoperimetric method, several versions of the bounded law of iterated logarithm for a sequence of independent Banach space valued random variables are developed and the upper limits for the non-random constant are given.

DOI : https://doi.org/10.1051/ps:2005002
Classification : 60F05,  60B12,  60F99
Mots clés : Banach space, bounded law of iterated logarithm, isoperimetric inequality, Rademacher series, self-normalizer
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     author = {Deng, Dianliang},
     title = {On the bounded laws of iterated logarithm in {Banach} space},
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     mrnumber = {2148959},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2005002/}
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Deng, Dianliang. On the bounded laws of iterated logarithm in Banach space. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 19-37. doi : 10.1051/ps:2005002. http://www.numdam.org/articles/10.1051/ps:2005002/

[1] A. De Acosta, Inequalities for B-valued random variables with application to the law of large numbers. Ann. Probab. 9 (1981) 157-161. | Zbl 0449.60002

[2] B. Von Bahr and C. Esseen, Inequalities for the rth absolute moments of a sum of random variables, 1r2. Ann. math. Statist. 36 (1965) 299-303. | Zbl 0134.36902

[3] X. Chen, On the law of iterated logarithm for independent Banach space valued random variables. Ann. Probab. 21 (1993) 1991-2011. | Zbl 0791.60005

[4] X. Chen, The Kolmogorov’s LIL of B-valued random elements and empirical processes. Acta Mathematica Sinica 36 (1993) 600-619. | Zbl 0785.60019

[5] Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martigales. Springer-Verlag, New York (1978). | MR 513230 | Zbl 0399.60001

[6] D. Deng, On the Self-normalized Bounded Laws of Iterated Logarithm in Banach Space. Stat. Prob. Lett. 19 (2003) 277-286. | Zbl 1113.60300

[7] U. Einmahl, Toward a general law of the iterated logarithm in Banach space. Ann. Probab. 21 (1993) 2012-2045. | Zbl 0790.60034

[8] E. Gine and J. Zinn, Some limit theorem for emperical processes. Ann. Probab. 12 (1984) 929-989. | Zbl 0553.60037

[9] A. Godbole, Self-normalized bounded laws of the iterated logarithm in Banach spaces, in Probability in Banach Spaces 8, R. Dudley, M. Hahn and J. Kuelbs Eds. Birkhäuser Progr. Probab. 30 (1992) 292-303. | Zbl 0787.60011

[10] P. Griffin and J. Kuelbs, Self-normalized laws of the iterated logarithm. Ann. Probab. 17 (1989) 1571-1601. | Zbl 0687.60033

[11] P. Griffin and J. Kuelbs, Some extensions of the LIL via self-normalizations. Ann. Probab. 19 (1991) 380-395. | Zbl 0722.60028

[12] M. Ledoux and M. Talagrand, Characterization of the law of the iterated logarithm in Babach spaces. Ann. Probab. 16 (1988) 1242-1264. | Zbl 0662.60008

[13] M. Ledoux and M. Talagrand, Some applications of isoperimetric methods to strong limit theorems for sums of independent random variables. Ann. Probab. 18 (1990) 754-789. | Zbl 0713.60005

[14] M. Ledoux and M. Talagrand, Probability in Banach Space. Springer-Verlag, Berlin (1991). | MR 1102015 | Zbl 0748.60004

[15] R. Wittmann, A general law of iterated logarithm. Z. Wahrsch. verw. Gebiete 68 (1985) 521-543. | Zbl 0547.60036

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