Geodesic
The bounded law of the iterated logarithm
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 3, pp. 522-537 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper deals with a sequence of identically distributed random variables with $\psi$-mixing and values in a type 2 Banach space whose norm meets certain conditions. The law of the iterated logarithm for this sequence is proved under optimal moment conditions.
Keywords: bounded law of iterated logarithm, $\psi$-mixing, type 2 Banach spaceю. 
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O. Sh. Sharipov. The bounded law of the iterated logarithm. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 3, pp. 522-537. http://geodesic.mathdoc.fr/item/TVP_2004_49_3_a5/

[1] Goodman V., Kuelbs J., Zinn J., “Some results on the LIL in Banach space with applications to weighted empirical processes”, Ann. Probab., 9:5 (1981), 713–752 | DOI | MR | Zbl

[2] de Acosta A., Kuelbs J., “Some results on the cluster sets $C(\{S_n/a_n\})$ and the LIL”, Ann. Probab., 11:1 (1983), 102–122 | DOI | MR | Zbl

[3] Ledoux M., Talagrand M., “Characterization of the law of the iterated logarithm in Banach spaces”, Ann. Probab., 16:3 (1988), 1242–1264 | DOI | MR | Zbl

[4] Kuelbs J., “A strong convergence theorem for Banach space valued random variables”, Ann. Probab., 4:5 (1976), 744–771 | DOI | MR | Zbl

[5] Kuelbs J., Philipp W., “Almost sure invariance principles for partial sums of mixing $B$-valued random variables”, Ann. Probab., 8:6 (1980), 1003–1036 | DOI | MR | Zbl

[6] Dehling H., Phillipp W., “Almost sure invariance principles for weakly dependent vector-valued random variables”, Ann. Probab., 10:3 (1982), 689–701 | DOI | MR | Zbl

[7] Berkes I., Philipp W., “Approximation theorems for independent and weakly dependent random vectors”, Ann. Probab., 7:1 (1979), 29–54 | DOI | MR | Zbl

[8] Phillipp W., “Almost sure invariance principles for sums of $B$-valued random variables”, Lecture Notes in Math., 709, 1979, 171–193 | MR

[9] Berger E., “An almost sure invariance principles for stationary ergodic sequences of Banach space valued random variables”, Probab. Theory Related. Fields, 84 (1990), 161–201 | DOI | MR | Zbl

[10] Einmahl U., “Stability results and strong invariance principles for partial sums of Banach space valued random variables”, Ann. Probab., 17:1 (1989), 333–352 | DOI | MR | Zbl

[11] Li D. L., Rao M. B., Wang X. C., “The law of the iterated logarithm for independent random variables with multidimensional indices”, Ann. Probab., 20:2 (1992), 660–674 | DOI | MR | Zbl

[12] Chen X., “On the law of the iterated logarithm for independent Banach space valued random variables”, Ann. Probab., 21:4 (1993), 1991–2011 | DOI | MR | Zbl

[13] Einmahl U., “Toward a general law of the iterated logarithm in Banach space”, Ann. Probab., 21:4 (1993), 2012–2045 | DOI | MR | Zbl

[14] Chen X., “On Strassen's law of the iterated logarithm in Banach space”, Ann. Probab., 22:2 (1994), 1026–1043 | DOI | MR | Zbl

[15] Einmahl U., “On the cluster set problem for the generalized law of the iterated logarithm in Euclidean space”, Ann. Probab., 23:2 (1995), 817–851 | DOI | MR | Zbl

[16] Li D., Tomkins R. J., “Compact laws of the iterated logarithm for $B$-valued random variables with two dimensional indices”, J. Theoret. Probab., 11:2 (1998), 443–459 | DOI | MR | Zbl

[17] Sharipov O. Sh., Predelnye teoremy dlya slabozavisimykh sluchainykh velichin so znacheniyami v prostranstve $l_p$, Avtoreferat diss. $\dots$ kand. fiz.-matem. nauk, Institut matematiki im. V. I. Romanovskogo AN Uzbekistana, Tashkent, 1993, 16 pp.

[18] Chen X., “The law of the iterated logarithm for $m$-dependent Banach space valued random variables”, J. Theoret. Probab., 10:3 (1997), 695–732 | DOI | MR

[19] Kuelbs J., “An inequality for the distribution of a sum of certain Banach space valued random variables”, Studia Math., 52 (1974), 69–87 | MR | Zbl

[20] Peligrad M., “Invariance principles for mixing sequences of random variables”, Ann. Probab., 10:4 (1982), 968–981 | DOI | MR | Zbl

[21] Sharipov O. Sh., “Zakon povtornogo logarifma dlya slabo zavisimykh sluchainykh velichin so znacheniyami v gilbertovom prostranstve”, Sib. matem. zhurnal, 44:6 (2003), 1407–1424 | MR | Zbl

[22] Sharipov O. Sh., “Strong limit theorems for mixing random variables with values in Hilbert space and their applications”, High dimensional probability, Progress in Probability, 55, eds. M. Marcus and J. Wellner, Birkhauser-Verlag, Basel: Switzerland, 2003, 153–174 | MR | Zbl

[23] Utev S. A., “Summy sluchainykh velichin s $\phi$-peremeshivaniem”, Trudy In-ta Matematiki SO AN RAN, 13, 1989, 78–100 | MR | Zbl

[24] Sharipov O. Sh., “O veroyatnostnoi kharakterizatsii banakhovykh prostranstv slabozavisimymi sluchainymi elementami”, Uzb. matem. zhurn., 1998, no. 2, 86–91 | MR

[25] Iosifescu M., Theodorescu R., Random Processes and Learning, Springer-Verlag, New York, 1969, 304 pp. | MR | Zbl

[26] de Acosta A., Giné E., “Convergence of moments and related functionals in the general central limit theorem in Banach spaces”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 48:2 (1979), 213–231 | DOI | MR | Zbl

[27] Bradley R. C., Introduction to strong mixing conditions,, Technical report, v. I. Department of Mathematics, Indiana University, March, Custom Publishing of I. U., Bloomington, Bloomington, 2002, 528 pp. | MR

[28] Shiryaev A. N., Veroyatnost, Nauka, M., 1980, 576 pp. | MR | Zbl

[29] Cohn H., “On a class of dependent random variables”, Rev. Roumaine Math. Pures Appl., 10 (1965), 1593–1606 | MR | Zbl