Towards Nikishin's Theorem on the Almost Sure Convergence of Rearrangements of Functional Series

Sh. Levental ; V. S. Mandrekar ; S. A. Chobanyan

Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 41-55.

Résumé

Necessary and sufficient conditions are found for the almost sure convergence of almost all simple rearrangements of a series of Banach space valued random variables. The results go back to Nikishin's well-known theorem on the existence of an almost surely convergent rearrangement of a numerical random series. An example is also given of a numerical random series with general term tending to zero almost surely such that this series converges in probability and any its rearrangement diverges almost surely.

Détail

Keywords: rearrangement of a series in a Banach space, almost sure convergence, ${\mathbf k}$-simple permutation, Nikishin's theorem.

Bibliographie

[1] “Problem 106 (S. Banach)”, The Scottish Book. Mathematics from Scottish Cafe, ed. R. D. Mauldin, Birkhäuser, Boston–Basel–Stuttgart, 1981 | MR

[2] W. Banaszczyk, “Steinitz theorem on rearrangement of series for nuclear spaces”, J. Reine Angew. Math., 403 (1990), 187–200 | DOI | MR | Zbl

[3] M.-J. Chasco, S. A. Chobanyan, “On rearrangements of series in locally convex spaces”, Mich. Math. J., 44:3 (1997), 607–617 | DOI | MR | Zbl

[4] S. A. Chobanyan, “Struktura mnozhestva summ uslovno skhodyaschegosya ryada v normirovannom prostranstve”, Matem. sb., 128(170):1(9) (1985), 50–65 | MR | Zbl

[5] S. A. Chobanyan, “Convergence a.s. of rearranged random series in Banach spaces and associated inequalities”, Probability in Banach Spaces, 9, Proc. of 9th International Conference on Probability in Banach Spaces (Sandbjerg, 1993), Prog. Probab., 35, Birkhäuser, Boston, MA, 1994, 3–29 | MR | Zbl

[6] S. Chobanyan, G. Giorgobiani, “A problem on rearrangements of summands in normed spaces and Rademacher sums”, Probability theory on vector spaces IV, Proc. of Conference (Lancut, 1987), Lecture Notes of Math., 1391, Springer-Verlag, Berlin, 33–46 | DOI | MR

[7] S. Chobanyan, G. Giorgobiani, “Almost sure permutational convergence of vector random series and Kolmogorov problem”, New Trends in Probability and Statistics, Proc. of 23rd Bakuriani Colloq. (Bakuriani 1990), 1, VSP, Utrecht, 1991, 93–105 | MR

[8] S. Chobanyan, S. Levental, V. Mandrekar, “Prokhorov blocks and strong law of large numbers under rearrangements”, J. Theoret. Probab., 17:3 (2004), 647–672 | DOI | MR | Zbl

[9] S. Chobanyan, V. Mandrekar, “Almost everywhere convergence and SLLN under rearrangements”, Stochastic processes and related topics, Trends Math., Birkhäuser, Boston, 1998, 25–34 | MR | Zbl

[10] S. Chobanyan, V. Mandrekar, “On Kolmogorov SLLN under rearrangements for “orthogonal” random variables in a B-space”, J. Theoret. Probab., 13:1 (2000), 135–139 | DOI | MR | Zbl

[11] V. P. Fonf, “Uslovno skhodyaschiesya ryady v ravnomerno vypuklom banakhovom prostranstve”, Matem. zametki, 11:2 (1972), 209–214 | MR | Zbl

[12] A. Garsia, “Existence of almost everywhere convergent rearrangements for Fourier series of $L_2$ functions”, Ann. of Math., 79 (1964), 623–629 | DOI | MR | Zbl

[13] A. Garsia, Topics in Almost Everywhere Convergence, Markham Publ. Co., Chicago, 1970 | MR | Zbl

[14] Z. G. Gorgadze, V. I. Tarieladze, S. A. Chobanyan, “Gaussovskie kovariatsii v banakhovykh podreshetkakh prostranstva $L_0(T,\Sigma, \nu)$”, Dokl. AN SSSR, 241:3 (1978), 528–531 | MR | Zbl

[15] M. I. Kadets, “Ob uslovno skhodyaschikhsya ryadakh v prostranstve $L^p$”, UMN, 9:1 (1954), 107–109 | MR | Zbl

[16] M. I. Kadets, V. M. Kadets, Series in Banach Spaces: Conditional and Unconditional Convergence, Birkhäuser, Boston, 1997 | MR | Zbl

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

[18] S. V. Konyagin, “O ravnomerno skhodyaschikhsya perestanovkakh trigonometricheskikh ryadov Fure”, Teoriya funktsii, SMFN, 25, RUDN, M., 2007, 80–87

[19] M. Ledoux, M. Talagrand, Probability in Banach Spaces. Isometry and Processes, Springer-Verlag, Berlin, 1991 | MR | Zbl

[20] S. Levental, “A maximal inequality for real numbers with application to exchangeable random variables”, TVP, 45:3 (2000), 615–621 | DOI | MR | Zbl

[21] V. Linde, A. Pich, “Otobrazheniya gaussovskikh tsilindricheskikh mer v banakhovykh prostranstvakh”, TVP, 19 (1974), 472–487 | MR | Zbl

[22] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, II, Springer-Verlag, Berlin–Heidelberg–New York, 1979 | MR | Zbl

[23] R. P. Maleev, “Ob uslovno skhodyaschikhsya ryadakh v nekotorykh banakhovykh reshetkakh”, C. R. Acad. Bulgare Sci., 32:8 (1979), 1015–1018 | MR | Zbl

[24] B. Maurey, “Type et cotype dans les espaces munis de structures locales inconditionelles”, Séminaire Maurey–Schwartz 1973–1974, Exp. 24, 25, Centre Math., École Polytech., Paris, 1974 | MR

[25] B. Maurey, G. Pisier, “Remarques sur l'exposé de P. Assouad "Espaces $p$-lisses, réarrangements"”, Séminaire Maurey–Schwartz 1974–1975, Annexe 1, Centre Math., École Polytech., Paris, 1975 | MR

[26] E. M. Nikishin, “O skhodyaschikhsya perestanovkakh funktsionalnykh ryadov”, Matem. zametki, 1:2 (1967), 129–136 | Zbl

[27] E. M. Nikishin, “O perestanovkakh ryadov v $L_p$”, Matem. zametki, 14:1 (1973), 31–38 | MR | Zbl

[28] D. V. Pecherskii, “Perestanovki ryadov v banakhovykh prostranstvakh i rasstanovki znakov”, Matem. sb., 135(177):1 (1988), 24–35 | MR

[29] Sz. Gy. Révész, “Rearrangement of Fourier series and Fourier series whose terms have random signs”, Acta Math. Hungar., 63:4 (1994), 395–402 | DOI | MR | Zbl

[30] E. Steinitz, “Bedingt konvergente Reihen und konvexe Systeme”, J. Reine Ang. Mathematik, 143 (1913), 128–175 ; 144 (1914), 1–40 ; 146 (1916), 1–52 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR

[31] S. L. Troyanskii, “Uslovnaya skhodimost ryadov i nekotorye F-prostranstva”, Teoriya funktsii i funkts. analiz i prilozh., 5 (1967), 102–107

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