Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T02:23:36.407Z Has data issue: false hasContentIssue false
  • English
  • Français

On the bounded laws of iterated logarithm inBanach space

Published online by Cambridge University Press:  15 November 2005

Dianliang Deng
Dianliang Deng*
Affiliation:
Department of Mathematics and Statistics, University of Regina, 3737 Wascana Parkway, Regina, SK, S4S 0A2 Canada; [email protected]
Get access

Abstract

In the present paper, by using theinequality due to Talagrand's isoperimetric method, severalversions of the bounded law of iterated logarithm for a sequenceof independent Banach space valued random variables are developedand the upper limits for the non-random constant are given.

Keywords

Banach spacebounded law of iteratedlogarithmisoperimetric inequalityrademacher seriesself-normalizer.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 19 - 37
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

de Acosta, A., Inequalities for B-valued random variables with application to the law of large numbers. Ann. Probab. 9 (1981) 157–161. CrossRef
von Bahr, B. and Esseen, C., Inequalities for the rth absolute moments of a sum of random variables, 1 ≤ r ≤ 2. Ann. math. Statist. 36 (1965) 299–303. CrossRef
Chen, X., On the law of iterated logarithm for independent Banach space valued random variables. Ann. Probab. 21 (1993) 1991–2011. CrossRef
Chen, X., The Kolmogorov's LIL of B-valued random elements and empirical processes. Acta Mathematica Sinica 36 (1993) 600–619.
Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martigales. Springer-Verlag, New York (1978).
Deng, D., On the Self-normalized Bounded Laws of Iterated Logarithm in Banach Space. Stat. Prob. Lett. 19 (2003) 277–286. CrossRef
Einmahl, U., Toward a general law of the iterated logarithm in Banach space. Ann. Probab. 21 (1993) 2012–2045. CrossRef
Gine, E. and Zinn, J., Some limit theorem for emperical processes. Ann. Probab. 12 (1984) 929–989. CrossRef
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.
Griffin, P. and Kuelbs, J., Self-normalized laws of the iterated logarithm. Ann. Probab. 17 (1989) 1571–1601. CrossRef
Griffin, P. and Kuelbs, J., Some extensions of the LIL via self-normalizations. Ann. Probab. 19 (1991) 380–395. CrossRef
Ledoux, M. and Talagrand, M., Characterization of the law of the iterated logarithm in Babach spaces. Ann. Probab. 16 (1988) 1242–1264. CrossRef
Ledoux, M. and Talagrand, M., Some applications of isoperimetric methods to strong limit theorems for sums of independent random variables. Ann. Probab. 18 (1990) 754–789. CrossRef
M. Ledoux and M. Talagrand, Probability in Banach Space. Springer-Verlag, Berlin (1991).
Wittmann, R., A general law of iterated logarithm. Z. Wahrsch. verw. Gebiete 68 (1985) 521–543. CrossRef