Geodesic
Systems of random variables equivalent in distribution to the Rademacher system and $\mathscr K$-closed representability of Banach couples
Sbornik. Mathematics, Tome 191 (2000) no. 6, pp. 779-807 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Necessary and sufficient conditions ensuring that one can select from a system $\{f_n\}_{n=1}^\infty$ of random variables on a probability space $(\Omega,\Sigma,\mathsf P)$ a subsystem $\{\varphi_i\}_{i=1}^\infty$ equivalent in distribution to the Rademacher system on $[0,1]$ are found. In particular, this is always possible if $\{f_n\}_{n=1}^\infty$ is a uniformly bounded orthonormal sequence. The main role in the proof is played by the connection (discovered in this paper) between the equivalence in distribution of random variables and the behaviour of the $L_p$-norms of the corresponding polynomials. An application of the results obtained to the study of the ${\mathscr K}$-closed representability of Banach couples is presented. 
@article{SM_2000_191_6_a0,
     author = {S. V. Astashkin},
     title = {Systems of random variables equivalent in distribution to {the~Rademacher} system and $\mathscr K$-closed representability of {Banach} couples},
     journal = {Sbornik. Mathematics},
     pages = {779--807},
     year = {2000},
     volume = {191},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_6_a0/}
}
TY  - JOUR
AU  - S. V. Astashkin
TI  - Systems of random variables equivalent in distribution to the Rademacher system and $\mathscr K$-closed representability of Banach couples
JO  - Sbornik. Mathematics
PY  - 2000
SP  - 779
EP  - 807
VL  - 191
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2000_191_6_a0/
LA  - en
ID  - SM_2000_191_6_a0
ER  - 
%0 Journal Article
%A S. V. Astashkin
%T Systems of random variables equivalent in distribution to the Rademacher system and $\mathscr K$-closed representability of Banach couples
%J Sbornik. Mathematics
%D 2000
%P 779-807
%V 191
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2000_191_6_a0/
%G en
%F SM_2000_191_6_a0
S. V. Astashkin. Systems of random variables equivalent in distribution to the Rademacher system and $\mathscr K$-closed representability of Banach couples. Sbornik. Mathematics, Tome 191 (2000) no. 6, pp. 779-807. http://geodesic.mathdoc.fr/item/SM_2000_191_6_a0/

[1] Astashkin S. V., “O vydelenii podsistem, “mazhoriruemykh” sistemoi Rademakhera”, Matem. zametki, 65:4 (1999), 483–495 | MR | Zbl

[2] Gaposhkin V. F., “Lakunarnye ryady i nezavisimye funktsii”, UMN, 21:6 (1966), 3–82 | MR | Zbl

[3] Khintchine A., “Über dyadische Brüche”, Math. Z., 18 (1923), 109–116 | DOI | MR | Zbl

[4] Zygmund A., “Sur les séries trigonométric lacunaires”, J. Lond. Math. Soc. (2), 5:2 (1930), 138–145 | DOI | Zbl

[5] Banach S., “Sur les séries lacunaires”, Bull. Int. Acad. Polon. Sci. A, 1933, no. 4/8, 149–154 | Zbl

[6] Kachmazh S., Shteingauz G., Teoriya ortogonalnykh ryadov, Fizmatgiz, M., 1958 | MR

[7] Zygmund A., “On lacunary trigonometric series”, Trans. Amer. Math. Soc., 34:3 (1932), 435–446 | DOI | MR | Zbl

[8] Kaczmarz S., Steinhaus G., “Le systéme orthogonal de M. Rademacher”, Studia Math., 2 (1930), 231–247 | Zbl

[9] Kashin B. S., “O nekotorykh svoistvakh prostranstva trigonometricheskikh polinomov s ravnomernoi normoi”, Tr. MIAN, 145, Nauka, M., 1980, 111–116 | MR | Zbl

[10] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[11] Berg I., Lefstrem I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[12] Montgomery-Smith S. J., “The distribution of Rademacher sums”, Proc. Amer. Math. Soc., 109:2 (1990), 517–522 | DOI | MR | Zbl

[13] Holmstedt T., “Interpolation of quasi-normed spaces”, Math. Scand., 26 (1970), 177–199 | MR | Zbl

[14] Gluskin E. D., Kwapien S., “Tail and moment estimates for sums of independent random variables with logarithmically concave tails”, Studia Math., 114:3 (1995), 303–309 | MR | Zbl

[15] Kakhan Zh.-P., Sluchainye funktsionalnye ryady, Mir, M., 1973 | MR | Zbl

[16] Jakubowski J., Kwapien S., “On multiplicative systems of functions”, Bull. Acad. Pol. Sci. Ser. Sci. Math., 27:9 (1979), 689–694 | MR | Zbl

[17] Pisier G., “Les inégalités de Khintchin–Kahane d'aprés C. Borell”, Semin. Geom. des Espaces de Banach, Ec. politech., Cent. Math., 1977–1978, no. 7, 1978, 1–14 | MR | Zbl

[18] Asmar N. H., Montgomery-Smith S., “On the distribution of Sidon series”, Ark. Mat., 31:1 (1993), 13–26 | DOI | MR | Zbl

[19] Belov A. S., Rodin V. A., “Normy lakunarnykh polinomov v funktsionalnykh prostranstvakh”, Matem. zametki, 51:3 (1992), 137–139 | MR | Zbl

[20] Rodin V. A., Semyonov E. M., “Rademacher series in symmetric spaces”, Anal. Math., 1:3 (1975), 207–222 | DOI | MR | Zbl

[21] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[22] Aleksich G., Problemy skhodimosti ortogonalnykh ryadov, IL, M., 1963 | MR

[23] Stechkin S. B., “Ob absolyutnoi skhodimosti ryadov Fure”, Izv. AN SSSR. Ser. matem., 20 (1956), 385–412 | MR | Zbl

[24] Astashkin S. V., “Ob interpolyatsii podprostranstv simmetrichnykh prostranstv, porozhdennykh sistemoi Rademakhera”, Izv. RAEN. Ser. MMMIU, 1:1 (1997), 18–35 | MR | Zbl

[25] Astashkin S. V., “O ryadakh po sisteme Rademakhera, “blizkikh” k $L_\infty$”, Funkts. analiz i ego prilozh., 32:3 (1998), 62–65 | MR | Zbl

[26] Lindenstrauss J., Tzafriri L., Classical Banach spaces, Springer-Verlag, Berlin, 1977 | MR