Skip to main content

The Comparison of Systems of Random Variables

  • Chapter
  • First Online:
The Rademacher System in Function Spaces
  • 387 Accesses

Abstract

The main purpose of this chapter is comparing systems of measurable functions (or r.v.’s) with the “model” Rademacher system. To clarify the meaning of the word “comparison” , we introduce the following definitions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Arias-de-Reyna, in Pointwise Convergence of Fourier Series. Lecture Notes in Math. vol. 1785 (Springer, 2002)

    Google Scholar 

  2. N.H. Asmar, S. Montgomery-Smith, On the distribution of Sidon series. Arkiv för Mat. 31, 13–26 (1993)

    Article  MathSciNet  Google Scholar 

  3. S.V. Astashkin, Extraction of subsystems “majorized” by the Rademacher system. Math. Notes 65, 407–417 (1999)

    Article  MathSciNet  Google Scholar 

  4. S.V. Astashkin, On the selection of subsystems equivalent in distribution to the Rademacher system. Math. Notes 66, 384–388 (1999)

    Article  MathSciNet  Google Scholar 

  5. S.V. Astashkin, Systems of random variables equivalent in distribution to the Rademacher system and \(\mathcal { K}\)-closed representability of Banach couples. Sbornik Math. 191, 779–807 (2000)

    Article  MathSciNet  Google Scholar 

  6. S.V. Astashkin, On the comparison of distribution functions of random variables. Math. Notes 87, 15–22 (2010)

    Article  MathSciNet  Google Scholar 

  7. A.S. Belov, V.A. Rodin, Norms of lacunary polynomials in functional spaces. Math. Notes 51, 318–320 (1992)

    Article  MathSciNet  Google Scholar 

  8. V.I. Bogachev, O.G. Smolyanov, Real and Functional Analysis (Springer, 2020)

    Google Scholar 

  9. J. Bourgain, M. Lewko, Sidonicity and variants of Kaczmarz’s problem. Ann. Inst. Fourier (Grenoble) 67, 1321–1352 (2017)

    Article  MathSciNet  Google Scholar 

  10. J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators (Cambridge University Press, Cambridge, 1995)

    Book  Google Scholar 

  11. K.E. Hare, R. (Xu) Yang, Sidon sets are proportionally Sidon with small Sidon constants. Can. Math. Bull. 62(4), 798–809 (2019)

    Google Scholar 

  12. T.P. Hÿtonen, J.M.A.M. van Neerven, M.C. Veraar, L.W. Weis, in Analysis in Banach Spaces, Vol. II: Probabilistic Methods and Operator Theory. A Series of Modern Surveys in Mathematics, vol. 67 (Springer, 2017)

    Google Scholar 

  13. J. Jakubowski, S. Kwapień, On multiplicative systems of functions. Bull. Acad. Pol. Sci. Ser. Sci. Math. 27, 689–694 (1979)

    MathSciNet  MATH  Google Scholar 

  14. S. Kwapień, W.A. Woyczyński, Random Series and Stochastic Integrals. Single and Multiple (Birkhäuser, Boston, 1992)

    Google Scholar 

  15. M. Ledoux, M. Talagrand, Probability in Banach Spaces (Springer, Berlin, 1991)

    Book  Google Scholar 

  16. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II, Function Spaces (Springer, Berlin, 1979).

    Book  Google Scholar 

  17. B. Maurey, in Type, Cotype and K-Convexity. Handbook of the Geometry of Banach Spaces, vol. 2 (North-Holland, Amsterdam, 2003), pp. 1299–1332

    Google Scholar 

  18. B. Maurey, G.-Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studia Math. 58, 45–90 (1976)

    Google Scholar 

  19. Y. Meyer, Algebraic Numbers and Harmonic Analysis (North-Holland, 1972)

    Google Scholar 

  20. W. Orlicz, Über unbedingte Konvergenz in Funktionräumen. Studia Math. 1, 83–85 (1930)

    Google Scholar 

  21. A. Pełczyński, Commensurate sequences of characters. Proc. Am. Math. Soc. 104, 525–531 (1988)

    Article  MathSciNet  Google Scholar 

  22. I. Pinelis, An asymptotically Gaussian bound on the Rademacher tails. Electron. J. Probab. 17(35), 1–22 (2012)

    MathSciNet  MATH  Google Scholar 

  23. G. Pisier, Les inégalités de Kahane–Khintchin d’après C. Borell, Séminaire sur la géometrie des espaces de Banach, Exposé N 7 (1977–1978), École Polytechnique, Palaiseau (in French)

    Google Scholar 

  24. G. Pisier, in De nouvelles caract é risations des ensembles de Sidon. Mathematical Anal. and Appl. Part B, vol. 7b. Adv. in Math. Suppl. Studies (Academic Press, New York, 1981), pp. 685–726

    Google Scholar 

  25. G. Pisier, Arithmetic characterizations of Sidon sets. Bull. Am. Math. Soc. 8, 87–89 (1983)

    Article  MathSciNet  Google Scholar 

  26. G. Pisier, On uniformly bounded orthonormal Sidon systems. Math. Res. Lett. 24, 893–932 (2017)

    Article  MathSciNet  Google Scholar 

  27. G. Pisier, Completely Sidon sets in discrete groups. arXiv:1706.03844v6 [math.OA] 6 Sep. 2018

    Google Scholar 

  28. V.A. Rodin, E.M. Semenov, Rademacher series in symmetric spaces. Anal. Math. 1, 207–222 (1975)

    Article  MathSciNet  Google Scholar 

  29. S.J. Szarek, in General Fourier Sums with Vector Coefficients and Analogs ofΛ(p)-Sets. Probability Theory and Harmonic Analysis (Cleveland, Ohio, 1983). Monogr. Textbooks. Pure Appl. Math., vol. 98 (Dekker, New York, 1986), pp. 195–208

    Google Scholar 

  30. N.N. Vakhania, V.1. Tarieladze, S.A. Chobanyan, in Probability Distributions on Banach Spaces. Math. Its Appl. (Soviet Series) (Kluwer, Holland, 1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Astashkin, S.V. (2020). The Comparison of Systems of Random Variables. In: The Rademacher System in Function Spaces. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-47890-2_7

Download citation

Publish with us

Policies and ethics