Some remarks on the ordinal strong law of large numbers
Author:
I. K. Matsak
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 72 (2006), 93-102
MSC (2000):
Primary 60B12
DOI:
https://doi.org/10.1090/S0094-9000-06-00667-3
Published electronically:
August 18, 2006
MathSciNet review:
2168139
Full-text PDF Free Access
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Abstract: We prove that the ordinal law of large numbers and the law of large numbers in the norm are equivalent for Banach lattices that do not contain uniformly the space $l_1^n$.
References
- Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015
- J. Hoffmann-Jørgensen, Probability in $B$-spaces, Matematisk Institut, Aarhus Universitet, Aarhus, 1977. Lecture Notes Series, No. 48. MR 0474447
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
- L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
- I. K. Matsak, The ordinal \lln in Banach lattices, Teor. Imovirnost. Matem. Statist. 62 (2000), 83–95; English transl. in Theory Probab. Math. Stat. 62 (2001), 89–102.
- William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- A. V. Buhvalov, A. I. Veksler, and V. A. Geĭler, Normed lattices, Mathematical analysis, Vol. 18 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1980, pp. 125–184 (Russian). MR 597904
- Ī. K. Matsak and A. M. Plīchko, On the maxima of independent random elements in a Banach functional lattice, Teor. Ĭmovīr. Mat. Stat. 61 (1999), 105–116 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 61 (2000), 109–120 (2001). MR 1866964
- Jon A. Wellner, A martingale inequality for the empirical process, Ann. Probability 5 (1977), no. 2, 303–308. MR 436296, DOI https://doi.org/10.1214/aop/1176995856
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability distributions on Banach spaces, Mathematics and its Applications (Soviet Series), vol. 14, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian and with a preface by Wojbor A. Woyczynski. MR 1435288
- A. V. Skorohod, Sluchaĭ nye protsessy s nezavisimymi prirashcheniyami, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0182056
- A. V. Skorohod, Sluchaĭ nye protsessy s nezavisimymi prirashcheniyami, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0182056
- Ī. K. Matsak, Estimates for the moments of the supremum of normed sums of independent random variables, Teor. Ĭmovīr. Mat. Stat. 67 (2002), 104–116 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 67 (2003), 115–128. MR 1956624
References
- M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, Berlin, 1991. MR 1102015 (93c:60001)
- J. Hoffmann-Jørgensen, Probability in $B$-spaces, Lect. Notes Series, vol. 48, Springer, Berlin, 1977. MR 0474447 (57:14087)
- J. Lindenstraus and L. Tzafriri, Classical Banach Spaces, vol. 2, Springer, Berlin, 1979. MR 0415253 (54:3344)
- L. V. Kantorovich and G. P. Akilov, Functional Analysis, “Nauka”, Moscow, 1984; English transl., Pergamon Press, New York, 1982. MR 0664597 (83h:46002)
- I. K. Matsak, The ordinal \lln in Banach lattices, Teor. Imovirnost. Matem. Statist. 62 (2000), 83–95; English transl. in Theory Probab. Math. Stat. 62 (2001), 89–102.
- W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, John Wiley & Sons, Inc., New York–London–Sydney, 1971. MR 0270403 (425292)
- G. Pizier, Sur les espaces qui ne contiennent pas $l^1_n$ uniformement, Séminaire Maurey–Schwartz 1973–74, Ecole Politechnique, Paris, 1974. MR 0270403 (42:5292)
- A. V. Bukhvalov, A. I. Veksler, and V. A. Geiler, Normed lattices, Itogi nauki. Mathematicheskii Analis, vol. 18, Akad. Nauk SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1980, pp. 125–184. (Russian) MR 0597904 (82b:46019)
- I. K. Matsak and A. M. Plichko, On the maxima of independent random elements in a Banach functional lattice, Teor. Imovirnost. Matem. Statist. 61 (1999), 105–116; English transl. in Theory Probab. Math. Stat. 61 (2000), 109–120. MR 1866964 (2002k:60020)
- J. A. Wellner, A martingale inequality for the empirical process, Ann. Probab. 5 (1977), no. 2, 303–308. MR 0436296 (55:9243)
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions on Banach Spaces, “Nauka”, Moscow, 1985; English transl., D. Reidel Publishing Co., Dordrecht, 1987. MR 1435288 (97k:60007)
- A. V. Skorokhod, Random Processes with Independent Increments, “Nauka”, Moscow, 1964; English transl., Kluwer Academic Publishers Group, Dordrecht, 1991. MR 0182056 (31:6280); MR 1155400 (93a:60114)
- J.-P. Kahane, Some Series of Functions, D. C. Heath and Company, Lexington, Massachusetts, 1968. MR 0182056 (31:6280)
- I. K. Matsak, Estimates for the moments of the supremum of normed sums of independent random variables, Teor. Imovirnost. Matem. Statist. 67 (2002), 104–116; English transl. in Theory Probab. Math. Stat. 67 (2003), 115–128. MR 1956624 (2004i:60061)
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Additional Information
I. K. Matsak
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
d.i.m.@ukrpost.net
Received by editor(s):
January 15, 2004
Published electronically:
August 18, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
