On the order law of the iterated logarithm

Matsak, I.

doi:10.1090/S0094-9000-04-00598-8

On the order law of the iterated logarithm

Author: I. K. Matsak
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 68 (2003).
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
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Abstract: We study the classical laws of the iterated logarithm due to Kolmogorov and Hartman-Wintner for random variables assuming values in Banach lattices.

1. V. V. Petrov, Summy nezavisimykh sluchaĭ nykh velichin, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0322927
2. 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
3. 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
4. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
5. L. V. Kantorovič and G. P. Akilov, Funktsional′nyĭ analiz v normirovannykh prostranstvakh, Gosudarstv. Izdat. Fis.-Mat. Lit., Moscow, 1959 (Russian). MR 0119071
6. Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824
7. 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.
8. 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
9. Mary Weiss, On the law of the iterated logarithm, J. Math. Mech. 8 (1959), 121–132. MR 0102853, https://doi.org/10.1512/iumj.1959.8.58008
10. A. I. Martikaĭnen, On the one-sided law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 30 (1985), no. 4, 694–705 (Russian). MR 816283
11. 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
12. S. A. Čobanjan and V. I. Tarieladze, A counter-example concerning CLT in Banach spaces, Probability theory on vector spaces (Proc. Conf., Trzebieszowice, 1977), Lecture Notes in Math., vol. 656, Springer, Berlin, 1978, pp. 25–30. MR 521018
13. I. K. Matsak and A. N. Plichko, A central limit theorem in a Banach space, Ukrain. Mat. Zh. 40 (1988), no. 2, 234–239, 272 (Russian); English transl., Ukrainian Math. J. 40 (1988), no. 2, 202–206. MR 941510, https://doi.org/10.1007/BF01056478
14. S. A. Rakov, Banach spaces in which Orlicz’s theorem is not valid, Mat. Zametki 14 (1973), 101–106 (Russian). MR 331038
15. N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, “Nauka”, Moscow, 1985 (Russian). MR 787803
16. V. A. Egorov, The law of the iterated logarithm, Teor. Verojatnost. i Primenen. 14 (1969), 722–729 (Russian, with English summary). MR 0266286