Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 

 

Ordinal law of the iterated logarithm in Banach lattices and some applications


Author: I. K. Matsak
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 77-91
MSC (2000): Primary 60B12
DOI: https://doi.org/10.1090/S0094-9000-07-00699-0
Published electronically: June 29, 2007
MathSciNet review: 2336780
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions are found for the ordinal law of the iterated logarithm in Banach lattices of type $ L^p$. As a corollary of our general results, we obtain a new law of the iterated logarithm for empirical processes in the spaces $ L^p(-\infty,\infty)$.


References [Enhancements On Off] (What's this?)

  • 1. A. Khintchin, Über einen Satz der Wahrscheinlichkeitsrechnung, Fund. Math. 6 (1924), 9-12.
  • 2. A. N. Kolmogorov, \cyr Teoriya veroyatnosteĭ i matematicheskaya statistika. Izbrannye trudy, “Nauka”, Moscow, 1986 (Russian). Edited by Yu. V. Prokhorov; With commentaries. MR 861120
    A. N. Kolmogorov, Selected works. Vol. II, Mathematics and its Applications (Soviet Series), vol. 26, Kluwer Academic Publishers Group, Dordrecht, 1992. Probability theory and mathematical statistics; With a preface by P. S. Aleksandrov; Translated from the Russian by G. Lindquist; Translation edited by A. N. Shiryayev [A. N. Shiryaev]. MR 1153022
  • 3. Philip Hartman and Aurel Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), 169–176. MR 0003497, https://doi.org/10.2307/2371287
  • 4. 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
  • 5. V. V. Petrov, \cyr Summy nezavisimykh sluchaĭnykh velichin., Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0322927
    V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. MR 0388499
  • 6. N. H. Bingham, Variants on the law of the iterated logarithm, Bull. London Math. Soc. 18 (1986), no. 5, 433–467. MR 847984, https://doi.org/10.1112/blms/18.5.433
  • 7. I. K. Matsak, On the law of the iterated logarithm in Banach lattices, Teor. Veroyatnost. i Primenen. 44 (1999), no. 4, 865–874 (Russian, with Russian summary); English transl., Theory Probab. Appl. 44 (2000), no. 4, 775–784. MR 1811138, https://doi.org/10.1137/S0040585X97977963
  • 8. Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824
  • 9. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
  • 10. 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
  • 11. Ī. K. Matsak, Mean 𝜓-deviation of a random element in a Banach lattice and some of its applications, Teor. Ĭmovīr. Mat. Stat. 60 (1999), 123–135 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 60 (2000), 137–149 (2001). MR 1826150
  • 12. 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
  • 13. V. Strassen, A converse to the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965/1966), 265–268. MR 0200965, https://doi.org/10.1007/BF00539114
  • 14. William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
  • 15. Gilles Pisier, Sur la loi du logarithme itéré dans les espaces de Banach, Probability in Banach spaces (Proc. First Internat. Conf., Oberwolfach, 1975), Springer, Berlin, 1976, pp. 203–210. Lecture Notes in Math., Vol. 526 (French). MR 0501237
  • 16. Ī. K. Matsak, A remark on the ordered law of large numbers, Teor. Ĭmovīr. Mat. Stat. 72 (2005), 84–92 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 72 (2006), 93–102. MR 2168139, https://doi.org/10.1090/S0094-9000-06-00667-3
  • 17. 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
  • 18. V. V. Jurinskiĭ, Exponential estimates for large deviations, Teor. Verojatnost. i Primenen. 19 (1974), 152–154 (Russian, with English summary). MR 0334298
  • 19. Bengt von Bahr and Carl-Gustav Esseen, Inequalities for the 𝑟th absolute moment of a sum of random variables, 1≤𝑟≤2, Ann. Math. Statist 36 (1965), 299–303. MR 0170407, https://doi.org/10.1214/aoms/1177700291
  • 20. M. Csörgő and P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 666546
  • 21. Peter Gänssler and Winfried Stute, Empirical processes: a survey of results for independent and identically distributed random variables, Ann. Probab. 7 (1979), no. 2, 193–243. MR 525051

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60B12

Retrieve articles in all journals with MSC (2000): 60B12


Additional Information

I. K. Matsak
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine
Address at time of publication: Kyiv National University for Technology and Design, Nemyrovych-Danchenko Street, 2, 01601, GSP, Kyiv, Ukraine
Email: m_i_k@ukr.net

DOI: https://doi.org/10.1090/S0094-9000-07-00699-0
Keywords: Law of the iterated logarithm, Banach lattices, empirical processes
Received by editor(s): May 7, 2004
Published electronically: June 29, 2007
Article copyright: © Copyright 2007 American Mathematical Society