Geodesic
Unconditional convergence of weakly sub gaussian series in Banach spaces
Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 2, pp. 295-318 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Characterizations of the class of Banach spaces isomorphing to the space $c_0$, as well as to the class of Banach spaces not containing $l_\infty^n$'s uniformly, are obtained in terms of almost surely unconditional convergence of weakly sub-Gaussian random series. Under almost surely unconditional convergence of random series, convergence of all permutations on the same set of full probability is understood. The questions of almost surely unconditional and weak absolute convergence in the spaces isomorphing to $c_0$ are investigated as well.
Keywords: Banach space with unconditional basis, Banach space not containing $l^n_\infty$'s uniformly, almost surely unconditional convergence, sub-Gaussian random variable, weakly sub-Gaussian random element, strongly sub-Gaussian random element, Gaussian random element. 
@article{TVP_2006_51_2_a2,
     author = {N. N. Vakhania and V. V. Kvaratskheliya},
     title = {Unconditional convergence of weakly sub gaussian series in {Banach} spaces},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {295--318},
     year = {2006},
     volume = {51},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a2/}
}
TY  - JOUR
AU  - N. N. Vakhania
AU  - V. V. Kvaratskheliya
TI  - Unconditional convergence of weakly sub gaussian series in Banach spaces
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2006
SP  - 295
EP  - 318
VL  - 51
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a2/
LA  - ru
ID  - TVP_2006_51_2_a2
ER  - 
%0 Journal Article
%A N. N. Vakhania
%A V. V. Kvaratskheliya
%T Unconditional convergence of weakly sub gaussian series in Banach spaces
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2006
%P 295-318
%V 51
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a2/
%G ru
%F TVP_2006_51_2_a2
N. N. Vakhania; V. V. Kvaratskheliya. Unconditional convergence of weakly sub gaussian series in Banach spaces. Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 2, pp. 295-318. http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a2/

[1] Buldygin V. V., Kozachenko Yu. V., “O subgaussovskikh sluchainykh velichinakh”, Ukrainskii matematicheskii zhurnal, 32:6 (1980), 723–730 | MR | Zbl

[2] Buldygin V. V., Kozachenko Yu. V., Metricheskie kharakteristiki sluchainykh velichin i protsessov, TViMS, Kiev, 1998, 290 pp. | Zbl

[3] Vakhaniya N. N., Kvaratskheliya V. V., Tarieladze V. I., “Slabo subgaussovskie sluchainye elementy v banakhovykh prostranstvakh”, Ukr. matem. zhurn., 57:9 (2005), 1187–1208 | MR | Zbl

[4] Vakhaniya N. N., Tarieladze V. I., Chobanyan S. A., Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985, 368 pp. | MR | Zbl

[5] Kvaratskheliya V. V., “Bezuslovnaya skhodimost gaussovskikh sluchainykh ryadov v banakhovykh prostranstvakh s bezuslovnym bazisom”, Teoriya veroyatn. i ee primen., 45:1 (2000), 178–182 | MR | Zbl

[6] Bessaga C., Pełczyński A., “On bases and unconditional convergence of series in Banach spaces”, Studia Math., 17:2 (1958), 151–164 | MR | Zbl

[7] Bagby R., Swartz Ch., “Projective tensor product of $l^p$-valued measures”, Mat. Slovaca, 25:3 (1975), 265–269 | MR | Zbl

[8] Chobanjan S. A., Tarieladze V. I., “Gaussian characterizations of certain Banach spaces”, J. Multivariate Anal., 7:1 (1977), 183–203 | DOI | MR | Zbl

[9] Dudley R. M., Pakula L., “A counter-example on the inner product of measures”, Indiana University Math. J., 21:9 (1972), 843–845 | DOI | MR | Zbl

[10] Fukuda R., “Exponential integrability of sub-Gaussian vectors”, Probab. Theory Related Fields, 85:4 (1990), 505–521 | DOI | MR | Zbl

[11] Kadec M. I., Pelczynski A., “Bases, lacunary sequences and complemented subspaces in the spaces $L_p$”, Studia Math., 21 (1961–1962), 161–176 | MR

[12] Kahane J. P., “Propriétés locales des fonctions à séries de Fourier aléatories”, Studia Math., 19:1 (1960), 1–25 | MR | Zbl

[13] Kvaratskhelia V. V., “On unconditional convergence of random series in Banach spaces”, Lect. Notes Math., 828, 1980, 162–166 | MR | Zbl

[14] Kvaratskhelia V., “Unconditional convergence of random series and geometry of Banach spaces”, Georgian Math. J., 7:1 (2000), 85–86 | MR

[15] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. I. Sequence Spaces, Springer-Verlag, Berlin, 1977, 188 pp. | MR

[16] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. II. Function Spaces, Springer-Verlag, Berlin, 1979, 243 pp. | MR

[17] Vakhania N. N., Probability Distributions on Linear Spaces, North Holland, New York–Oxford, 1981, 123 pp. | MR | Zbl

[18] Vakhania N., “On subgaussian random vectors in normed spaces”, Bull. Georgian Acad. Sci., 163:1 (2001), 8–10 | MR | Zbl

[19] Vakhania N., Kvaratskhelia V., “Absolute and unconditional convergence in $l_1$”, Bull. Georgian Acad. Sci., 160:2 (1999), 201–203 | MR | Zbl

[20] Vakhania N., Kvaratskhelia V., “On a criterion for unconditional convergence of Hadamard series in the spaces $l_p$, $1\le p\infty$”, Bull. Georgian Acad. Sci., 162:2 (2000), 199–202 | MR | Zbl

[21] Vakhania N., Kvaratskhelia V., “Convergence of Sylvester series in Banach space $l_p$, $1\le p\infty$”, Bull. Georgian Acad. Sci., 164:1 (2001), 7–9 | MR | Zbl