Convergence of series of scalar- and vector-valued random variables and a subsequence principle in 𝐿₂

Dilworth, S. J.

doi:10.1090/S0002-9947-1987-0879579-X

Convergence of series of scalar- and vector-valued random variables and a subsequence principle in $L_ 2$
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by S. J. Dilworth

Trans. Amer. Math. Soc. 301 (1987), 375-384

DOI: https://doi.org/10.1090/S0002-9947-1987-0879579-X

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Abstract:

Let $({d_n})_{n = 1}^\infty$ be a martingale difference sequence in ${L_0}(X)$, where $X$ is a uniformly convex Banach space. We investigate a necessary condition for convergence of the series $\sum {_{n = 1}^\infty {a_n}{d_n}}$. We also prove a related subsequence principle for the convergence of a series of square-integrable scalar random variables.

References

D. J. Aldous and D. H. Fremlin, Colacunary sequences in $L$-spaces, Studia Math. 71 (1981/82), no. 3, 297–304. MR 667318, DOI 10.4064/sm-71-3-297-304
G. Alexits, Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York-Oxford-Paris, 1961. Translated from the German by I. Földer. MR 0218827
S. D. Chatterji, A general strong law, Invent. Math. 9 (1969/70), 235–245. MR 266276, DOI 10.1007/BF01404326
Y. S. Chow, Martingale extensions of a theorem of Marcinkiewicz and Zygmund, Ann. Math. Statist. 40 (1969), 427–433. MR 239642, DOI 10.1214/aoms/1177697707
Yuan Shih Chow and Henry Teicher, Probability theory, Springer-Verlag, New York-Heidelberg, 1978. Independence, interchangeability, martingales. MR 513230, DOI 10.1007/978-1-4684-0062-5
S. J. Dilworth, Universal noncompact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space, Israel J. Math. 52 (1985), no. 1-2, 15–27. MR 815597, DOI 10.1007/BF02776075
Leonard E. Dor, Some inequalities for martingales and applications to the study of $L_{1}$, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 135–148. MR 591980, DOI 10.1017/S0305004100058023
Tadeusz Figiel and Gilles Pisier, Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses, C. R. Acad. Sci. Paris Sér. A 279 (1974), 611–614 (French). MR 358295
Richard F. Gundy, The martingale version of a theorem of Marcinkiewicz and Zygmund, Ann. Math. Statist. 38 (1967), 725–734. MR 215353, DOI 10.1214/aoms/1177698865
Robert C. James, Some self-dual properties of normed linear spaces, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N.J., 1972, pp. 159–175. MR 0454600
M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/62), 161–176. MR 152879, DOI 10.4064/sm-21-2-161-176
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9

Sur les fonctions independants

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Terry R. McConnell, Stable-bounded subsets of $L^\alpha$, and sample unboundedness of symmetric stable processes, J. Funct. Anal. 60 (1985), no. 2, 265–279. MR 777239, DOI 10.1016/0022-1236(85)90053-9
J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed. MR 0402915
Gilles Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), no. 3-4, 326–350. MR 394135, DOI 10.1007/BF02760337
P. Révész, On a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 16 (1965), 311–318 (English, with Russian summary). MR 185647, DOI 10.1007/BF01904839
Pál Révész, The laws of large numbers, Probability and Mathematical Statistics, Vol. 4, Academic Press, New York-London, 1968. MR 0245079
Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
William F. Stout, Almost sure convergence, Probability and Mathematical Statistics, Vol. 24, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0455094
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

Trigonometric series