The Annals of Probability

Best constants in Rosenthal-type inequalities and the Kruglov operator

S. V. Astashkin and F. A. Sukochev

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Abstract

Let X be a symmetric Banach function space on [0, 1] with the Kruglov property, and let f={fk}k=1n, n≥1 be an arbitrary sequence of independent random variables in X. This paper presents sharp estimates in the deterministic characterization of the quantities

‖∑k=1nfkX, ‖(∑k=1n|fk|p)1/pX,  1≤p<∞,

in terms of the sum of disjoint copies of individual terms of f. Our method is novel and based on the important recent advances in the study of the Kruglov property through an operator approach made earlier by the authors. In particular, we discover that the sharp constants in the characterization above are equivalent to the norm of the Kruglov operator in X.

Article information

Source
Ann. Probab., Volume 38, Number 5 (2010), 1986-2008.

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1282053778

Digital Object Identifier
doi:10.1214/10-AOP529

Mathematical Reviews number (MathSciNet)
MR2722792

Zentralblatt MATH identifier
1211.46008

Subjects
Primary: 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 60G50: Sums of independent random variables; random walks

Keywords
Kruglov property Rosenthal inequality symmetric function spaces

Citation

Astashkin, S. V.; Sukochev, F. A. Best constants in Rosenthal-type inequalities and the Kruglov operator. Ann. Probab. 38 (2010), no. 5, 1986--2008. doi:10.1214/10-AOP529. https://projecteuclid.org/euclid.aop/1282053778


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References

  • [1] Astashkin, S. V. (2005). Extrapolation functors on a family of scales generated by the real interpolation method. Sibirsk. Mat. Zh. 46 264–289.
  • [2] Astashkin, S. V. and Sukochev, F. A. (2005). Series of independent random variables in rearrangement invariant spaces: An operator approach. Israel J. Math. 145 125–156.
  • [3] Astashkin, S. V. and Sukochev, F. A. (2004). Comparison of sums of independent and disjoint functions in symmetric spaces. Mat. Zametki 76 483–489.
  • [4] Astashkin, S. V. and Sukochev, F. A. (2008). Series of independent mean zero random variables in rearrangement-invariant spaces having the Kruglov property. J. Math. Sci. 148 795–809.
  • [5] Braverman, M. S. (1994). Independent Random Variables and Rearrangement Invariant Spaces. London Mathematical Society Lecture Note Series 194. Cambridge Univ. Press, Cambridge.
  • [6] Bukhvalov, A. V. (1987). Interpolation of linear operators in spaces of vector functions and with a mixed norm. Sibirsk. Mat. Zh. 28 37–51.
  • [7] Burkholder, D. L. (1979). A sharp inequality for martingale transforms. Ann. Probab. 7 858–863.
  • [8] Calderón, A. P. (1964). Intermediate spaces and interpolation, the complex method. Studia Math. 24 113–190.
  • [9] Creekmore, J. (1981). Type and cotype in Lorentz Lpq spaces. Nederl. Akad. Wetensch. Indag. Math. 43 145–152.
  • [10] Hitczenko, P. and Montgomery-Smith, S. (2001). Measuring the magnitude of sums of independent random variables. Ann. Probab. 29 447–466.
  • [11] Johnson, W. B., Maurey, B., Schechtman, G. and Tzafriri, L. (1979). Symmetric structures in Banach spaces. Mem. Amer. Math. Soc. 19 v+298.
  • [12] Johnson, W. B. and Schechtman, G. (1989). Sums of independent random variables in rearrangement invariant function spaces. Ann. Probab. 17 789–808.
  • [13] Johnson, W. B., Schechtman, G. and Zinn, J. (1985). Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13 234–253.
  • [14] Junge, M. (2006). The optimal order for the p-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates. Positivity 10 201–230.
  • [15] Kruglov, V. M. (1970). A remark on the theory of infinitely divisible laws. Teor. Veroyatn. Primen. 15 330–336.
  • [16] Kreĭn, S. G., Petunīn, Y. Ī. and Semënov, E. M. (1982). Interpolation of Linear Operators. Translations of Mathematical Monographs 54. Amer. Math. Soc., Providence, RI.
  • [17] Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
  • [18] Latała, R. (1997). Estimation of moments of sums of independent real random variables. Ann. Probab. 25 1502–1513.
  • [19] Lozanovskiĭ, G. J. (1972). A remark on a certain interpolation theorem of Calderón. Funktsional. Anal. i Prilozhen. 6 89–90.
  • [20] Lindenstrauss, J. and Tzafriri, L. (1979). Classical Banach Spaces. II. Function Spaces. Ergebnisse der Mathematik und Ihrer Grenzgebiete [Results in Mathematics and Related Areas] 97. Springer, Berlin.
  • [21] Montgomery-Smith, S. (2002). Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables. Israel J. Math. 131 51–60.
  • [22] Prohorov, Y. V. (1958). Strong stability of sums and infinitely divisible laws. Teor. Veroyatn. Primen. 3 153–165.
  • [23] Rosenthal, H. P. (1970). On the subspaces of Lp (p>2) spanned by sequences of independent random variables. Israel J. Math. 8 273–303.