Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Symmetric polynomials of random variables attracted to an infinitely divisible law
Download PDF
Download PDF
  • Published: December 1986

Symmetric polynomials of random variables attracted to an infinitely divisible law

  • Florin Avram1 &
  • Murad S. Taqqu1 nAff2 

Probability Theory and Related Fields volume 71, pages 491–500 (1986)Cite this article

  • 101 Accesses

  • 14 Citations

  • Metrics details

Summary

LetX i,n′ i=1, ...n be a triangular array in the domain of attraction of a Lévy processX(t). Then the symmetric polynomial of orderk in theX i,n′

$$\sum\limits_{1 \leqq i_1< i_2< \ldots< i_k \leqq [nt]} {X_{i_1 ,n} X_{i_2 ,n} \ldots X_{i_k ,n} ,}$$

converges weakly to thek th order multiple integral with respect toX(t). Also, sums of powers of theX i,n converge jointly to the variations of the processX(t).

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

    Google Scholar 

  2. Dynkin, E.B., Mandelbaum, A.: Symmetric statistics, Poisson point processes and multiple Wiener integrals. Ann. Stat.11, 739–745 (1983)

    Google Scholar 

  3. Denker, M., Grillenberger, Chr., Keller, G.: A note on invariance principles for von Mises' statistics. Metrika (to appear 1985.)

  4. Feinsilver, P.J.: Special functions, probability semigroups and Hamiltonian flows. Lect. Notes in Math.696. Berlin-Heidelberg-New York: Springer 1978

    Google Scholar 

  5. Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Reading, Mass: Addison-Wesley 1954

    Google Scholar 

  6. Holley, R., Strock, D.W.: Central limit phenomena of various interacting systems. Ann. Math.110, 333–393 (1979)

    Google Scholar 

  7. Itô, K.: Stochastic Processes. Lecture Notes Series16, Mathematisk Institut, Aarhus Universitet (1969)

  8. Mandelbaum, A., Taqqu, M.S.: Invariance principle for symmetric statistics. Ann. of Statistics12, 483–496 (1984)

    Google Scholar 

  9. Meyer, P.-A.: Un cours sur les intégrales stochastiques. Sem. Probl. X. Lect. Notes Math.511, 245–400. Berlin-Heidelberg-New York: Springer 1976

    Google Scholar 

  10. Rvaceva, E.L.: On domains of attraction of multi-dimensional distributions. Selected Transl. in Probab. v.2, 1962, 183–205 (1954)

    Google Scholar 

  11. Rubin, H., Vitale, R.A.: Asymptotic distribution of symmetric statistics. Ann. of Statistics8, 165–170 (1980)

    Google Scholar 

Download references

Author information

Author notes
  1. Murad S. Taqqu

    Present address: Department of Mathematics, Boston University, 02215, Boston, MA, USA

Authors and Affiliations

  1. School of Operations Research, Cornell University, 14853, Ithaca, NY, USA

    Florin Avram & Murad S. Taqqu

Authors
  1. Florin Avram
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Murad S. Taqqu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The second author is supported in part by NSF grant ECS 84-08524 at Cornell University.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Avram, F., Taqqu, M.S. Symmetric polynomials of random variables attracted to an infinitely divisible law. Probab. Th. Rel. Fields 71, 491–500 (1986). https://doi.org/10.1007/BF00699038

Download citation

  • Received: 15 March 1984

  • Revised: 10 June 1985

  • Issue Date: December 1986

  • DOI: https://doi.org/10.1007/BF00699038

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Symmetric Polynomial
  • Triangular Array
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature