On the order law of the iterated logarithm
Author:
I. K. Matsak
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 68 (2004), 93-101
MSC (2000):
Primary 60B12
DOI:
https://doi.org/10.1090/S0094-9000-04-00598-8
Published electronically:
May 24, 2004
MathSciNet review:
2000398
Full-text PDF Free Access
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Additional Information
Abstract: We study the classical laws of the iterated logarithm due to Kolmogorov and Hartman–Wintner for random variables assuming values in Banach lattices.
References
- V. V. Petrov, Sums of Independent Random Variables, “Nauka", Moscow, 1972; English transl., Springer-Verlag, Berlin, 1975. MR 48:1288
- N. H. Bingham, Variants of the law of the iterated logarithm, Bull. London Math. Soc. 18 (1986), 433–467. MR 87k:60087
- M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, Berlin, 1991. MR 93c:60001
- J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, 1979. MR 58:17766
- L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Fizmatgiz, Moscow, 1959; English transl., Mcmillan, New York, 1964. MR 22:9837
- K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965. MR 31:5054
- I. K. Matsak Mean $\psi$-deviation of a random element in a Banach lattice and its applications, Teor. Veroyatnost. i Mat. Statist. 60 (1999), 129–141; English transl. in Theory Probab. Math. Statist. 60 (2000), 137–149.
- ---, On the law of the iterated logarithm in Banach lattices, Teor. Veroyatnost. i Primenen. 44 (1999), no. 4, 865–874; English transl. in Theory Probab. Appl. 44 (2000), no. 4. MR 2003a:60008
- M. Weiss, On the law of the iterated logarithm, J. Math. Mech. 8 (1959), 121–132. MR 21:1639
- A. I. Martikainen, On the one-sided law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 30 (1985), no. 4, 694–705; English transl. in Theory Probab. Appl. 30 (1986), no. 4. MR 87a:60038
- W. Feller An Introduction to Probability Theory and Its Applications, vol. 2, Wiley, New York, 1971. MR 42:5292
- S. A. Chobanyan and V. I. Tarieladze, A counterexample concerning the CLT in Banach spaces, Lect. Notes Math. 656 (1978), 25–30. MR 80d:60015
- I. K. Matsak and A. N. Plichko, Central limit theorem in a Banach space, Ukrain. Mat. Zh. 40 (1988), no. 2, 234–239; English transl. in Ukrainian Math. J. 40 (1989). MR 89g:60015
- S. A. Rakov, On Banach spaces for which an Orlicz theorem does not hold, Mat. Zametki 14 (1973), 101–106; English transl. in Math. Notes 14 (1974). MR 48:9373
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions on banach Spaces, “Nauka", Moscow, 1985; English transl., Kluwer, Dordrecht, 1987. MR 86j:60014
- V. A. Egorov, On the law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 14 (1969), no. 4, 722–729; English transl. in Theory Probab. Appl. 14 (1970), no. 4. MR 42:1193
References
- V. V. Petrov, Sums of Independent Random Variables, “Nauka", Moscow, 1972; English transl., Springer-Verlag, Berlin, 1975. MR 48:1288
- N. H. Bingham, Variants of the law of the iterated logarithm, Bull. London Math. Soc. 18 (1986), 433–467. MR 87k:60087
- M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, Berlin, 1991. MR 93c:60001
- J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, 1979. MR 58:17766
- L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Fizmatgiz, Moscow, 1959; English transl., Mcmillan, New York, 1964. MR 22:9837
- K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965. MR 31:5054
- I. K. Matsak Mean $\psi$-deviation of a random element in a Banach lattice and its applications, Teor. Veroyatnost. i Mat. Statist. 60 (1999), 129–141; English transl. in Theory Probab. Math. Statist. 60 (2000), 137–149.
- ---, On the law of the iterated logarithm in Banach lattices, Teor. Veroyatnost. i Primenen. 44 (1999), no. 4, 865–874; English transl. in Theory Probab. Appl. 44 (2000), no. 4. MR 2003a:60008
- M. Weiss, On the law of the iterated logarithm, J. Math. Mech. 8 (1959), 121–132. MR 21:1639
- A. I. Martikainen, On the one-sided law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 30 (1985), no. 4, 694–705; English transl. in Theory Probab. Appl. 30 (1986), no. 4. MR 87a:60038
- W. Feller An Introduction to Probability Theory and Its Applications, vol. 2, Wiley, New York, 1971. MR 42:5292
- S. A. Chobanyan and V. I. Tarieladze, A counterexample concerning the CLT in Banach spaces, Lect. Notes Math. 656 (1978), 25–30. MR 80d:60015
- I. K. Matsak and A. N. Plichko, Central limit theorem in a Banach space, Ukrain. Mat. Zh. 40 (1988), no. 2, 234–239; English transl. in Ukrainian Math. J. 40 (1989). MR 89g:60015
- S. A. Rakov, On Banach spaces for which an Orlicz theorem does not hold, Mat. Zametki 14 (1973), 101–106; English transl. in Math. Notes 14 (1974). MR 48:9373
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions on banach Spaces, “Nauka", Moscow, 1985; English transl., Kluwer, Dordrecht, 1987. MR 86j:60014
- V. A. Egorov, On the law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 14 (1969), no. 4, 722–729; English transl. in Theory Probab. Appl. 14 (1970), no. 4. MR 42:1193
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Additional Information
I. K. Matsak
Affiliation:
Kyiv State University for Technology and Design, Nemirovich-Danchenko Street 2, Kyiv 02011, Ukraine
Received by editor(s):
September 1, 2000
Published electronically:
May 24, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
