Skip to main content
SpringerLink
Log in

On the Almost Sure Summability of B-Valued Random Variables

  • Chapter
Book cover Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference

Part of the book series: Progress in Probability ((PRPR,volume 30))

  • 602 Accesses

Abstract

Since the origin of Probability in Banach spaces, many papers have been devoted to the following problem: “If \(X=(X_{k})_{k\geq1}\) is a sequence of scalar valued random variables (r.v.), which are independent and identically distributed, then the strong law of large numbers (SLLN) (under various forms: Kolmogorov, Erdös-Hsu-Robbins, Marcinkiewicz-Zygmund,…), the central-limit theorem (CLT) or the law of the iterated logarithm (LIL) hold for X as soon as a suitable integrability condition ℑ(∣X 1∣) is fulfilled. If now the X k take their values in a real separable Banach space (B, ∥ ∥) — equipped with its Borel σ-field B — does the condition ℑ(∥X 1∥) also characterize the SLLN, the CLT or the LIL for X?”

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

Access this chapter

Log in via an institution
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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. De Acosta (A.). — Strong exponential integrability of sums of independent B-valued random vectors, Probab. Math. Statist., t. 1, 1980, p. 133–150.

    MathSciNet  MATH  Google Scholar 

  2. De Acosta (A.). — Inequalities for B-valued random vectors with applications to the strong law of large numbers, Ann. Probab., t. 9, 1981, p. 157–161.

    Article  MathSciNet  MATH  Google Scholar 

  3. Alt (J.C.). — Une forme générale de la loi forte des grands nombres pour des variables aléatoires vectorielles, Probability Theory on Vector Spaces TV — Lancut 1987, p. 1–15, Lecture Notes in Math. 1391, Berlin, Springer, 1989.

    Google Scholar 

  4. Araujo (A.), Gine (E.). — The central limit theorem for real and Banach space valued random variables. — New York, Wiley, 1980.

    Google Scholar 

  5. Azlarov (T.A.), Volodin (N.A.), — Laws of large numbers for identically distributed Banach space valued random variables, Theory Probab. Appl., t. 26, 1981, p. 573–580.

    Article  MathSciNet  Google Scholar 

  6. Bingham (N.H.). — Extensions of the strong law, Adv. in Appl. Probab. Supplement (G.E.H. Reuter Festschrift, ed. D.G. Kendall), 1986, p. 27–36.

    Google Scholar 

  7. Bingham (N.H.), Maejima (M.). — Summability methods and almost sure convergence, Z. Wahr. verw. Geb., t. 68, 1985, p. 383–392.

    Article  MathSciNet  MATH  Google Scholar 

  8. Bingham (N.H.), Stadtmuller (U.). — Jakimovski methods and almost sure convergence, Disorder in Physical Systems, J.M. Hammersley Festschrift, ed. G.R. Grimmett, D.J.A. Welsh, p. 5–17, Oxford Univ. Press, 1990.

    Google Scholar 

  9. Bingham (N.H.), Tenenbaum (G.). — Riesz and Valiron means and fractional moments, Math. Proc. Cambridge Philos. Soc., t. 99, 1986, p. 143–149.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bozorgnia (A.), Bhaskara Rao (M.). — On summability methods and limit theorems for Banach space valued random variables, Bull. Inst. Math. Acad. Sinica, t. 7, 1979, p. 1–6.

    MathSciNet  MATH  Google Scholar 

  11. Chow (Y.S.). — Delayed sums and Borel summability of independent, identically distributed random variables, Bull. Inst. Math. Acad. Sinica, t. 1, 1973, P. 207–220.

    MathSciNet  MATH  Google Scholar 

  12. Chow (Y.S.), Lai (T.L.). — Limiting behavior of weighted sums of independent random variables, Ann. Probab., t. 1, 1973, p. 810–824.

    Article  MathSciNet  MATH  Google Scholar 

  13. Chow (Y.S.), Teicher (H.). — Probability theory. — Berlin, Springer, 1978.

    Book  MATH  Google Scholar 

  14. Deniel (Y.), Derriennic (Y.). — Sur la convergence presque sûre, au sens de Cesaro d’ordre α, 0 < α < 1, de variables aléatoires indépendantes et identiquement distribuées, Probab. Theory Related Fields, t. 79, 1988, p. 629–636.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hardy (G.H.). — Divergent Series. — Oxford, Clarendon Press, 1949.

    MATH  Google Scholar 

  16. Hardy (G.H.), Littlewood (J.E.). — Theorems concerning the summability of series by Borel’s exponential method, Rendiconti Circolo Matematico de Palermo 41, p. 36–53, 1916, and in Collected works of G.H. Hardy, vol. VI, p. 609-628, Clarendon Press, Oxford, 1974.

    Article  MATH  Google Scholar 

  17. Heinkel (B.). — An infinite dimensional law of large numbers, in Cesaro’s sense, J. Theoret. Probab., t. 3, 1990, p. 533–546.

    Article  MathSciNet  MATH  Google Scholar 

  18. Heinkel (B.). — On Valiron means of B-valued random variables, Preprint, 1991.

    Google Scholar 

  19. Hoffmann-jørgensen (J.), Pisier (G.). — The law of large numbers and the central limit theorem in Banach spaces, Ann. Probab., t. 4, 1976, p. 587–599.

    Article  Google Scholar 

  20. Kuelbs (J.), Zinn (J.). — Some stability results for vector valued random variables, Ann. Probab., t. 7, 1979, p. 75–84.

    Article  MathSciNet  MATH  Google Scholar 

  21. Lai (T.L.). — Summability methods for independent, identically distributed random variables, Proc. Amer. Math. Soc., t. 45, 1974, p. 253–261.

    MathSciNet  MATH  Google Scholar 

  22. Lai (T.L.). — Convergence rates in the strong law of large numbers for random variables taking values in Banach spaces, Bull. Inst. Math. Acad. Sinica, t. 2, 1974, p. 67–85.

    MathSciNet  MATH  Google Scholar 

  23. Ledoux (M.), Talagrand (M.). — Characterization of the law of the iterated logarithm in Banach spaces, Ann. Probab., t. 16, 1988, p. 1242–1264.

    Article  MathSciNet  MATH  Google Scholar 

  24. Ledoux (M.), Talagrand (M.). — Comparison theorems, random geometry and some limit theorems for empirical processes, Ann. Probab., t. 17, 1989, p. 596–631.

    Article  MathSciNet  MATH  Google Scholar 

  25. Ledoux (M.), Talagrand (M.). — Some applications of isoperimetric methods to strong limit theorems for sums of independent random variables, Ann. Probab., t. 18, 1990, p. 754–789.

    Article  MathSciNet  MATH  Google Scholar 

  26. Ledoux (M.), Talagrand (M.). — Probability in Banach spaces. — Berlin, Springer, 1991.

    MATH  Google Scholar 

  27. Lorentz (G.G.). — Borel and Banach properties of methods of summation, Duke Math J., t. 22, 1955, p. 129–141.

    Article  MathSciNet  MATH  Google Scholar 

  28. Martikainen (A.L). — Regular methods of summing random terms, Theory Probab. Appl., t. 30, 1985, p. 9–18.

    Article  Google Scholar 

  29. Mikosch (T.), Norvaisa (R.). — Limit theorems for methods of summation of independent random variables, I, Lithuanian Math. J., t. 27, 1987, p. 83–93.

    Article  MATH  Google Scholar 

  30. Mikosch (T.), Norvaisa (R.). — Limit theorems for methods of summation of independent random variables, II, Lithuanian Math. J., t. 27, 1987, p. 128–144.

    Article  MATH  Google Scholar 

  31. Mourier (E.). — Eléments aléatoires dans un espace de Banach, Ann. Inst. H. Poincaré, t. 13, 1953, p. 159–244.

    MathSciNet  Google Scholar 

  32. Pisier (G.). — Sur la loi du logarithme itéré dans les espaces de Banach, Probability in Banach spaces 1, Oberwolfach 1975, p. 203–210, Lecture Notes in Math 526, Berlin, Springer, 1976.

    Google Scholar 

  33. Stout (W.F.). — Almost sure convergence. — New York, Academic Press, 1974.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématique, Université de Strasbourg 1, 7, rue René Descartes, 67084, Strasbourg, France

    Bernard Heinkel

Authors
  1. Bernard Heinkel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Mathematics, MIT, Cambridge, MA, 02139, USA

    Richard M. Dudley

  2. Department of Mathematics, Tufts University, Medford, MA, 02178, USA

    Marjorie G. Hahn

  3. Dept. of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    James Kuelbs

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Science+Business Media New York

About this chapter

Cite this chapter

Heinkel, B. (1992). On the Almost Sure Summability of B-Valued Random Variables. In: Dudley, R.M., Hahn, M.G., Kuelbs, J. (eds) Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Progress in Probability, vol 30. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0367-4_22

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0367-4_22

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6728-7

  • Online ISBN: 978-1-4612-0367-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation