Almost Everywhere Convergence and SLLN Under Rearrangements

  • Sergei Chobanyan
  • V. Mandrekar
Part of the Trends in Mathematics book series (TM)


The almost everywhere (a.e.) convergence of trigonometric Fourier series for L 2(0, 1) functions was conjectured by Luzin (1922) and was partially solved by Kolmogorov and Silvestrov in (1925). The full solution was given by Carleson (1966). In the work of Garsia (1964), (see Garsia (1970)) almost everywhere convergence of a rearrangement of series of orthogonal functions was initiated using the so-called Garsia Inequality (GI). His convergence result was generalized by Nikishin (1967) who removed the assumption of orthogonality. We prove an inequality which generalizes GI using the technique introduced in Chobanyan (1990) (for other references see Chobanyan (1994)). As a consequence of this inequality we derive GI and a generalization of Nikishin’s result to a series of Banach space valued random variables.


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  1. [1]
    Beck A., Giesy D., Warren P. (1975). Recent developments in the theory of strong law of large numbers for vector valued random variables. Th. Prob. Appl. 20 106–133.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Beck A., Warren P. (1972). Weak orthogonality. Pac. J. Math. 41, 1–11.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Carleson L. (1966). On convergence and growth of partial sums of Fourier series. Acta Math. (Uppsala). 116, 135–157.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Chobanyan S. (1990). On some inequalities related to permutations of summands in a normed space, Preprint, Muskhelishvili Inst. Comp. Math.}, Georgian Acad. Sc. 1–21.Google Scholar
  5. [5]
    Chobanyan S. (1994). Convergence a.s. of rearranged series in Banach spaces and associated inequalities. Prob. in Banach spaces 9 (eds. J. Hoffman-Jorgensen, J. Kuelbs, and M.B. Marcus), Birkhäuser Boston.Google Scholar
  6. [6]
    Csorgo S., Tandori K., Totek W. (1983). On strong law of large numbers for pairwise independent random variables. Acta Math. Hung. 42, 319–330.CrossRefGoogle Scholar
  7. [7]
    Garsia A. (1964). Existence of almost everywhere convergent rearrangement for Fourier series of L 2 functions. Ann. Math. 79, 623–629.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Garsia A. (1970). Topics in almost everywhere convergence, Markham Publishing Company, Chicago.zbMATHGoogle Scholar
  9. [9]
    Kolmogorov A.N., Silvestrov G.S. (1925). Sur la convergence des series de Fourier. Comptes Rend. Acad. Sci Paris. 178, 303–305.Google Scholar
  10. [10]
    Luzin N.N. (1951). Integral and Trigonometric Series. 2nd ed. Gostekhiz-dat. Moscow (Russian).Google Scholar
  11. [11]
    Moricz F.(1983). On the Cesaro means of orthogonal sequences of random variables. Ann. Prob. 11, 827–832.Google Scholar
  12. [12]
    Nikishin E.M.(1967). On convergent rearrangements of functional series. Matem. Zametki 1, 126–136. (Russian).Google Scholar
  13. [13]
    Tandori K. (1982). Bemerkung zum Gesetzt der grossen Zahlen. Acta Math. Hung. 39 361–362.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Sergei Chobanyan
    • 1
  • V. Mandrekar
    • 1
  1. 1.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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