Electronic Journal of Probability

A Functional Combinatorial Central Limit Theorem

Andrew Barbour and Svante Janson

Full-text: Open access

Abstract

The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process. Distance is measured by comparison of expectations of smooth functionals of the processes, and the argument is by way of Stein's method. The pre-limiting process is then shown, under weak conditions, to converge to a Gaussian limit process. The theorem is used to describe the shape of random permutation tableaux.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 81, 2352-2370.

Dates
Accepted: 30 October 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819542

Digital Object Identifier
doi:10.1214/EJP.v14-709

Mathematical Reviews number (MathSciNet)
MR2556014

Zentralblatt MATH identifier
1193.60010

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60F17: Functional limit theorems; invariance principles 62E20: Asymptotic distribution theory 05E10: Combinatorial aspects of representation theory [See also 20C30]

Keywords
Gaussian process combinatorial central limit theorem permutation tableau Stein's method

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Barbour, Andrew; Janson, Svante. A Functional Combinatorial Central Limit Theorem. Electron. J. Probab. 14 (2009), paper no. 81, 2352--2370. doi:10.1214/EJP.v14-709. https://projecteuclid.org/euclid.ejp/1464819542


Export citation

References

  • R. J. Adler & J. E. Taylor (2007) Random Fields and Geometry. Springer, New York.
  • A. D. Barbour (1990) Stein's method for diffusion approximation. Prob. Theory Rel. Fields 84, 297–322.
  • P. Billingsley (1968) Convergence of Probability Measures. Wiley, New York.
  • E. Bolthausen (1984) An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrscheinlichkeit verw. Geb. 66, 379–386.
  • P. Hitczenko & S. Janson (2009) Asymptotic normality of statistics on permutation tableaux. Preprint, arXiv:0904.1222
  • W. Hoeffding (1951) A combinatorial central limit theorem. Ann. Math. Stat. 22, 558–566.
  • M. R. Leadbetter, G. Lindgren & H. Rootzén (1983) Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • G. R. Shorack & J. A. Wellner (1986) Empirical Processes with Applications to Statistics. Wiley, New York.
  • E. Steingrímsson & L. K. Williams (2007) Permutation tableaux and permutation patterns. J. Comb. Theory, Ser. A 114, 211–234.