×

zbMATH — the first resource for mathematics

On the bounded laws of iterated logarithm in Banach space. (English) Zbl 1136.60314
Summary: 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.
MSC:
60F05 Central limit and other weak theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F99 Limit theorems in probability theory
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] A. de Acosta , Inequalities for \(B\)-valued random variables with application to the law of large numbers . Ann. Probab. 9 ( 1981 ) 157 - 161 . Article | Zbl 0449.60002 · Zbl 0449.60002 · doi:10.1214/aop/1176994517 · minidml.mathdoc.fr
[2] B. von Bahr and C. Esseen , Inequalities for the \(r\)th absolute moments of a sum of random variables, \(1\leq r\leq 2\) . Ann. math. Statist. 36 ( 1965 ) 299 - 303 . Article | Zbl 0134.36902 · Zbl 0134.36902 · doi:10.1214/aoms/1177700291 · minidml.mathdoc.fr
[3] X. Chen , On the law of iterated logarithm for independent Banach space valued random variables . Ann. Probab. 21 ( 1993 ) 1991 - 2011 . Article | Zbl 0791.60005 · Zbl 0791.60005 · doi:10.1214/aop/1176989008 · minidml.mathdoc.fr
[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 · 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 · 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 · Zbl 1113.60300 · doi:10.1016/S0167-7152(03)00172-X
[7] U. Einmahl , Toward a general law of the iterated logarithm in Banach space . Ann. Probab. 21 ( 1993 ) 2012 - 2045 . Article | Zbl 0790.60034 · Zbl 0790.60034 · doi:10.1214/aop/1176989009 · minidml.mathdoc.fr
[8] E. Gine and J. Zinn , Some limit theorem for emperical processes . Ann. Probab. 12 ( 1984 ) 929 - 989 . Article | Zbl 0553.60037 · Zbl 0553.60037 · doi:10.1214/aop/1176993138 · minidml.mathdoc.fr
[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 · Zbl 0787.60011
[10] P. Griffin and J. Kuelbs , Self-normalized laws of the iterated logarithm . Ann. Probab. 17 ( 1989 ) 1571 - 1601 . Article | Zbl 0687.60033 · Zbl 0687.60033 · doi:10.1214/aop/1176991175 · minidml.mathdoc.fr
[11] P. Griffin and J. Kuelbs , Some extensions of the LIL via self-normalizations . Ann. Probab. 19 ( 1991 ) 380 - 395 . Article | Zbl 0722.60028 · Zbl 0722.60028 · doi:10.1214/aop/1176990551 · minidml.mathdoc.fr
[12] M. Ledoux and M. Talagrand , Characterization of the law of the iterated logarithm in Babach spaces . Ann. Probab. 16 ( 1988 ) 1242 - 1264 . Article | Zbl 0662.60008 · Zbl 0662.60008 · doi:10.1214/aop/1176991688 · minidml.mathdoc.fr
[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 . Article | Zbl 0713.60005 · Zbl 0713.60005 · doi:10.1214/aop/1176990857 · minidml.mathdoc.fr
[14] M. Ledoux and M. Talagrand , Probability in Banach Space . Springer-Verlag, Berlin ( 1991 ). MR 1102015 | Zbl 0748.60004 · Zbl 0748.60004
[15] R. Wittmann , A general law of iterated logarithm . Z. Wahrsch. verw. Gebiete 68 ( 1985 ) 521 - 543 . Zbl 0547.60036 · Zbl 0547.60036 · doi:10.1007/BF00535343
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.