Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T15:23:20.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Zero Biasing and Jack Measures

Published online by Cambridge University Press:  20 June 2011

JASON FULMAN  and
LARRY GOLDSTEIN
JASON FULMAN
Affiliation:
University of Southern California, Los Angeles, CA 90089-2532, USA (e-mail: [email protected], [email protected])
LARRY GOLDSTEIN
Affiliation:
University of Southern California, Los Angeles, CA 90089-2532, USA (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The tools of zero biasing are adapted to yield a general result suitable for analysing the behaviour of certain growth processes. The main theorem is applied to prove a central limit theorem, with explicit error terms in the L1 metric, for a natural statistic of the Jack measure on partitions.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 753 - 762
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aldous, D. and Diaconis, P. (1999) Longest increasing subsequences: From patience sorting to the Baik–Deift–Johansson theorem. Bull. Amer. Math. Soc. (NS) 36 413–432.CrossRefGoogle Scholar
[2]Borodin, A., Okounkov, A. and Olshanski, G. (2000) Asymptotics of Plancherel measure for symmetric groups. J. Amer. Math. Soc. 13 481–515.Google Scholar
[3]Borodin, A. and Olshanski, G. (2005) Z-measures on partitions and their scaling limits. Europ. J. Combin. 26 795–834.CrossRefGoogle Scholar
[4]Chen, L., Goldstein, L. and Shao, Q. (2010) Normal Approximation by Stein's Method, Springer.Google Scholar
[5]Deift, P. (2000) Integrable systems and combinatorial theory. Notices Amer. Math. Soc. 47 631–640.Google Scholar
[6]Diaconis, P. and Holmes, S. (2002) Random walks on trees and matchings. Electron. J. Probab. 7.CrossRefGoogle Scholar
[7]Fulman, J. (2004) Stein's method, Jack measure, and the Metropolis algorithm. J. Combin. Theory Ser. A 108 275–296.Google Scholar
[8]Fulman, J. (2005) Stein's method and Plancherel measure of the symmetric group. Trans. Amer. Math. Soc. 357 555–570.CrossRefGoogle Scholar
[9]Fulman, J. (2006) An inductive proof of the Berry–Esseen theorem for character ratios. Ann. Combin. 10 319–332.CrossRefGoogle Scholar
[10]Fulman, J. (2006) Martingales and character ratios. Trans. Amer. Math. Soc. 358 4533–4552.Google Scholar
[11]Fulman, J. and Goldstein, L. (2010) Zero biasing and growth processes. arXiv:1010.4759Google Scholar
[12]Goldstein, L. (2005) Berry–Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42 661–683.CrossRefGoogle Scholar
[13]Goldstein, L. (2007) L 1 bounds in normal approximation. Ann. Probab. 35 1888–1930.CrossRefGoogle Scholar
[14]Goldstein, L. (2010) Bounds on the constant in the mean central limit theorem. Ann. Probab. 38 1672–1689.CrossRefGoogle Scholar
[15]Goldstein, L. (2004) Normal approximation for hierarchical sequences. Ann. Appl. Probab. 14 1950–1969.Google Scholar
[16]Goldstein, L. and Reinert, G. (1997) Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.CrossRefGoogle Scholar
[17]Hora, A. (1998) Central limit theorem for the adjacency operators on the infinite symmetric group. Comm. Math. Phys. 195 405–416.CrossRefGoogle Scholar
[18]Hora, A. and Obata, N. (2007) Quantum Probability and Spectral Analysis of Graphs, Theoretical and Mathematical Physics, Springer.Google Scholar
[19]Ivanov, V. and Olshanski, G. (2002) Kerov's central limit theorem for the Plancherel measure on Young diagrams. In Symmetric Functions 2001: Surveys of Developments and Perspectives, Kluwer.Google Scholar
[20]Johansson, K. (2001) Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. 153 259–296.Google Scholar
[21]Kerov, S. V. (1993) Gaussian limit for the Plancherel measure of the symmetric group. Compt. Rend. Acad. Sci. Paris Ser. I 316 303–308.Google Scholar
[22]Kerov, S. V. (2000) Anisotropic Young diagrams and Jack symmetric functions. Funct. Anal. Appl. 34 41–51.CrossRefGoogle Scholar
[23]Kerov, S. V. (1996) The boundary of Young lattice and random Young tableaux. In Formal Power Series and Algebraic Combinatorics: New Brunswick, NJ, 1994, Vol. 24 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., AMS.Google Scholar
[24]Okounkov, A. (2000) Random matrices and random permutations. Internat. Math. Res. Notices 20 1043–1095.CrossRefGoogle Scholar
[25]Okounkov, A. (2005) The uses of random partitions. In XIVth International Congress on Mathematical Physics, World Scientific, pp. 379–403.Google Scholar
[26]Rinott, Y. and Rotar, V (1997) On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics. Ann. Appl. Probab. 7 1080–1105.Google Scholar
[27]Shao, Q. and Su, Z. (2006) The Berry–Esseen bound for character ratios. Proc. Amer. Math. Soc. 134 2153–2159.Google Scholar
[28]Sniady, P. (2006) Gaussian fluctuations of characters of symmetric groups and of Young diagrams. Probab. Theory Rel. Fields 136 263–297.CrossRefGoogle Scholar
[29]Stein, C. (1972) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Prob. Vol. 2, University of California Press, pp. 586–602.Google Scholar
[30]Stein, C. (1986) Approximate Computation of Expectations, IMS.CrossRefGoogle Scholar