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
Approximation theorems for random permanents and associated stochastic processes
Download PDF
Download PDF
  • Published: 11 November 2004

Approximation theorems for random permanents and associated stochastic processes

  • Grzegorz A. Rempała1,3 &
  • Jacek Wesołowski2 

Probability Theory and Related Fields volume 131, pages 442–458 (2005)Cite this article

  • 104 Accesses

  • 2 Citations

  • Metrics details

Abstract.

The limit theorems for certain stochastic processes generated by permanents of random matrices of independent columns with exchangeable components are established. The results are based on the martingale decomposition of a random permanent function similar to the one known for U-statistics and on relating the components of this decomposition to some multiple stochastic integrals.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Billingsley, P.: Convergence of probability measures. John Wiley & Sons Inc., New York, 2nd edn. A Wiley-Interscience Publication, 1999

  2. Brualdi, R.A., Ryser, H.J.: Combinatorial matrix theory, vol. 39 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1991

  3. Chow, Y.S., Teicher, H.: Probab. theory. Springer-Verlag, New York, 1978. Independence, interchangeability, martingales

  4. Coelho, M.P., Duffner, M.A.: On the relation between the determinant and the permanent on symmetric matrices. Linear Multilinear Algebra 51 (2), 127–136 (2003)

    Google Scholar 

  5. Forbert, H., Marx, D.: Calculation of the permanent of a sparse positive matrix. Comput. Phys. Comm. 150 (3), 267–273 (2003)

    Google Scholar 

  6. Jakubowski, A., Mémin, J., Pagès, G.: Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod. Probab. Theory Relat Fields 81 (1), 111–137 (1989)

    Google Scholar 

  7. Janson, S.: The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph. Combin. Probab. Comput. 3 (1), 97–126 (1994)

    Google Scholar 

  8. Korolyuk, V.S., Borovskikh, Yu.V.: Random permanents and symmetric statistics. In Probab. theory Math. Stat. (Kiev, 1991), pp. 176–187; World Sci. Publishing, River Edge, NJ, 1992

  9. Kurtz, T.G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (3), 1035–1070 (1991)

    Google Scholar 

  10. Lee, A.J.: U-statistics.Theory and practice, vol. 110 of Statistics: Textbooks and Monographs. Marcel Dekker Inc., New York, 1990

  11. Móri, T.F., Székely, G.J.: Asymptotic behaviour of symmetric polynomial statistics. Ann. Probab. 10 (1), 124–131 (1982)

    Google Scholar 

  12. Pate, T.H.: The best lower bound for the permanent of a correlation matrix of rank two. Linear Multilinear Algebra 51 (3), 263–278 (2003)

    Google Scholar 

  13. Rempała, G., Wesołowski, J.: Asymptotic behavior of random permanents. Statist. Probab. Lett. 45, 149–158 (1999)

    Google Scholar 

  14. Rempała, G., Wesołowski, J.: Central limit theorems for random permanents with correlation structure. J. Theoret. Probab. 15 (1), 63–76 (2002)

    Google Scholar 

  15. Rempała, G., Wesołowski, J.: Limit theorems for random permanents with exchangeable structure. J. Multivariate Analysis. To appear

  16. Ryser, H.J.: Combinatorial mathematics. Published by The Mathematical Association of America, 1963

  17. Székely, G.J., Szeidl, L.: On the limit distribution of random permanents. In: Exploring stochastic laws, VSP, Utrecht, 1995, pp. 443–455

  18. Székely, G.J.: A limit theorem for elementary symmetric polynomials of independent random variables. Z. Wahrsch. Verw. Gebiete 59 (3), 355–359 (1982)

    Google Scholar 

  19. Taylor, R.L., Daffer, P.Z., Patterson, R.F.: Limit theorems for sums of exchangeable random variables. Rowman & Allanheld Publishers, Totowa, N.J., 1985

  20. Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8 (2), 189–201 (1979)

    Google Scholar 

  21. van Es, A.J., Helmers, R.: Elementary symmetric polynomials of increasing order. Probab. Theory Relat. Fields 80 (1), 21–35 (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Its Applications, University of Minnesota, USA

    Grzegorz A. Rempała

  2. Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Polland

    Jacek Wesołowski

  3. Department of Mathematics, University of Louisville, USA

    Grzegorz A. Rempała

Authors
  1. Grzegorz A. Rempała
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacek Wesołowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Mathematics Subject Classification (2000): Primary 60F17, 62G20; Secondary 15A15, 15A52

Acknowledgement A significant part of this work was completed when the first author was visiting the Center for Mathematical Sciences at the University of Wisconsin-Madison. He would like to express his gratitude to the Center and its Acting Director, Prof. Thomas G. Kurtz, for their hospitality. Thanks are also due to the first author’s current host, the Institute for Mathematics and Its Applications at the University of Minnesota. Finally, both authors graciously acknowledge the comments of an anonymous referee on an earlier version of this paper.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rempała, G., Wesołowski, J. Approximation theorems for random permanents and associated stochastic processes. Probab. Theory Relat. Fields 131, 442–458 (2005). https://doi.org/10.1007/s00440-004-0380-9

Download citation

  • Received: 19 November 2003

  • Revised: 19 March 2004

  • Published: 11 November 2004

  • Issue Date: March 2005

  • DOI: https://doi.org/10.1007/s00440-004-0380-9

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

Key words or phrases:

  • Random permanent
  • Permanent process
  • Orthogonal decomposition
  • Invariance principle
  • Donsker’s theorem
  • Elementary symmetric polynomial process
  • Multiple stochastic integral
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