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Extraction of subsystems “Majorized” by the rademacher system

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Abstract

In this paper it is proved that from any uniformly bounded orthonormal system {f n} ∞n=1 of random variables defined on the probability space (Ω, ε, P), one can extract a subsystem {fni} Emphasis>=1/∞i majorized in distribution by the Rademacher system on [0, 1]. This means that {

$$P\left\{ {\omega \in \Omega :\left| {\sum\limits_{i = 1}^m {a_i f_{n_i } } \left( \omega \right)} \right| > z} \right\} \leqslant C\left| {\left\{ {t \in \left[ {0,1} \right]:\left| {\sum\limits_{i = 1}^m {a_i r_i \left( t \right)} } \right| > \frac{z}{C}} \right\}} \right|.$$

}, whereC>0 is independent of m∈N, ai∈N (i=1,…,m) andz>0.

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References

  1. S. Kaczmarz and H. Steinhaus,Theorie der Orthogonalreihen, Warsaw-Lvov (1935).

  2. C. G. Krein, Yu. I. Petunin, and E. M. Semenov,Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  3. J. Bergh and J. Löfström,Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg-New York (1976).

    MATH  Google Scholar 

  4. S. V. Astashkin, “On the interpolation of subspaces of symmetric spaces generated by the Rademacher system,”Izv. RAEN. Ser. MMMIU 1, No. 1, 18–35 (1997).

    MATH  Google Scholar 

  5. S. V. Astashkin, “Series in the Rademacher system that are “close to”L ∞,”Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.] (to appear).

  6. S. Montgomery-Smith, “The distribution of Rademacher sums,”Proc. Amer. Math. Soc.,109, No. 2, 517–522 (1990).

    Article  MATH  Google Scholar 

  7. T. Holmstedt, “Interpolation of quasi-normed spaces,”Math. Scand.,26, 177–199 (1970).

    MATH  Google Scholar 

  8. S. J. Szarek, “On the best constants in the Khinchine inequality,”Studia Math.,58, 197–208 (1976).

    MATH  Google Scholar 

  9. N. H. Asmar and S. Montgomery-Smith, “On the distribution of Sidon series,”Ark. Mat.,31, No. 1, 13–26 (1993).

    Article  MATH  Google Scholar 

  10. Ya. B. Rutitskii, “On some classes of measurable functions,”Uspekhi Mat. Nauk [Russian Math. Surveys],20, No. 4, 205–208 (1965).

    Google Scholar 

  11. A. Zygmund,Trigonometric Series, Vol. 1, Cambridge Univ. Press, Cambridge (1959).

    MATH  Google Scholar 

  12. R. Elliott,Stochastic Calculus and Applications, Springer-Verlag, Heidelberg (1983).

    Google Scholar 

  13. J. Jakubowski and S. Kwapien, “On multiplicative systems of functions,”Bull. Acad. Polon. Sci. Ser. Sci. Math.,27, No. 9, 689–694 (1979).

    MATH  Google Scholar 

  14. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Vol. 2, Springer, Berlin (1979).

    MATH  Google Scholar 

  15. V. F. Gaposhkin, “Convergence and limiting theorems for subsequences of random values,”Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.],17, No. 3, 401–423 (1972).

    Google Scholar 

  16. V. F. Gaposhkin, “Gap series and independent functions,”Uspekhi Mat. Nauk [Russian Math. Surveys],21 No. 6, 3–82 (1966).

    Google Scholar 

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Authors and Affiliations

  1. Samara State University, Samara, USSR

    S. V. Astashkin

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  1. S. V. Astashkin
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Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 483–495, April, 1999.

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Astashkin, S.V. Extraction of subsystems “Majorized” by the rademacher system. Math Notes 65, 407–417 (1999). https://doi.org/10.1007/BF02675354

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