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Characterization of LIL behavior in Banach space
Author(s):
Uwe
Einmahl;
Deli
Li
Journal:
Trans. Amer. Math. Soc.
360
(2008),
6677-6693.
MSC (2000):
Primary 60B12, 60F15;
Secondary 60G50, 60J15
Posted:
July 24, 2008
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Abstract:
In a recent paper by the authors a general result characterizing two-sided LIL behavior for real valued random variables has been established. In this paper we look at the corresponding problem in the Banach space setting. We show that there are analogous results in this more general setting. In particular, we provide a necessary and sufficient condition for LIL behavior with respect to sequences of the form  , where  is from a suitable subclass of the positive, non-decreasing slowly varying functions. To prove these results we have to use a different method. One of our main tools is an improved Fuk-Nagaev type inequality in Banach space which should be of independent interest.
References:
-
- 1.
- A. de Acosta, J. Kuelbs and M. Ledoux An inequality for the law of the iterated logarithm. In: Probability in Banach spaces 4. Lecture Notes in Mathematics, vol. 990, Springer, Berlin, Heidelberg, 1983, pp. 1-29. MR 707506 (85a:60016)
- 2.
- O. Bousquet A Bennett concentration inequality and its application to suprema of empirical processes, C. R. Math. Acad. Sci. Paris 334 (2002), 495-500. MR 1890640 (2003f:60039)
- 3.
- U. Einmahl Toward a general law of the iterated logarithm in Banach space. Ann. Probab. 21 (1993), 2012-2045. MR 1245299 (95d:60010)
- 4.
- U. Einmahl A generalization of Strassen's functional LIL. J. Theoret. Probab. 20 (2007), 901-915. MR 2359061
- 5.
- U. Einmahl and D. Li Some results on two-sided LIL behavior. Ann. Probab. 33, (2005) 1601-1624. MR 2150200 (2006b:60049)
- 6.
- W. Feller An extension of the law of the iterated logarithm to variables without variance. J. Math. Mech. 18, (1968), 343-355. MR 0233399 (38:1721)
- 7.
- M. Klass Toward a universal law of the iterated logarithm I. Z. Wahrsch. Verw. Gebiete 36, (1976), 165-178. MR 0415742 (54:3822)
- 8.
- M. Klass Toward a universal law of the iterated logarithm II. Z. Wahrsch. Verw. Gebiete 39, (1977), 151-165. MR 0448502 (56:6808)
- 9.
- T. Klein and E. Rio Concentration around the mean for maxima of empirical processes. Ann. Probab. 33, (2005), 1060-1077. MR 2135312 (2006c:60022)
- 10.
- J. Kuelbs The LIL when
 is in the domain of attraction of a Gaussian law. Ann. Probab. 13, (1985), 825-859. MR 799424 (86m:60020) - 11.
- M. Ledoux On Talagrand's deviation inequality for product measures. ESAIM Probab. Statist. 1, (1996), 63-87. MR 1399224 (97j:60005)
- 12.
- M. Ledoux and M. Talagrand Characterization of the law of the iterated logarithm in Banach space. Ann. Probab. 16, (1988), 1242-1264. MR 942766 (89i:60016)
- 13.
- M. Ledoux and M. Talagrand, M. Probability in Banach Spaces. Springer, Berlin, 1991. MR 1102015 (93c:60001)
- 14.
- P. Massart About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28, (2000), 863-884. MR 1782276 (2001m:60038)
- 15.
- V.V. Petrov Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford, 1995. MR 1353441 (96h:60048)
- 16.
- W. Pruitt General one-sided laws of the iterated logarithm. Ann. Probab. 9, (1981), 1-48. MR 606797 (82k:60066)
- 17.
- E. Rio Une inégalité de Bennet pour les maxima de processus empiriques. Ann. Inst. H. Poincaré Probab. Statist. 38, (2002), 1053-1057. MR 1955352 (2005e:60047)
- 18.
- M. Talagrand Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22, (1994), 28-76. MR 1258865 (95a:60064)
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Additional Information:
Uwe
Einmahl
Affiliation:
Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
Email:
ueinmahl@vub.ac.be
Deli
Li
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1
Email:
dli@lakeheadu.ca
DOI:
10.1090/S0002-9947-08-04522-4
PII:
S 0002-9947(08)04522-4
Keywords:
Law of the iterated logarithm,
LIL behavior,
Banach space,
regularly varying function,
sums of i.i.d. random variables,
exponential inequalities
Received by editor(s):
October 16, 2006
Received by editor(s) in revised form:
April 1, 2007
Posted:
July 24, 2008
Additional Notes:
The first author’s research was supported in part by an FWO Vlaanderen grant.
The second author’s research was supported in part by an NSERC Canada grant
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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