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On Statistics of Permutations Chosen From the Ewens Distribution

Published online by Cambridge University Press:  22 August 2014

TATJANA BAKSHAJEVA  and
EUGENIJUS MANSTAVIČIUS
TATJANA BAKSHAJEVA
Affiliation:
Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania (e-mail: [email protected])
EUGENIJUS MANSTAVIČIUS
Affiliation:
Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, LT-08663 Vilnius, Lithuania (e-mail: [email protected]
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Abstract

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles function is analysed in more detail. The results can be applied to statistics defined on random permutation matrices.

Keywords

Primary 60C05Secondary 05A1620P05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014 , pp. 889 - 913
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Arous, G. B. and Dang, K. (2011) On fluctuations of eigenvalues of random permutation matrices. arXiv:1106.2108v1Google Scholar
[2]Arratia, R.Barbour, A. D. and Tavaré, S. (2003) Logarithmic Combinatorial Structures: A Proba-bilistic Approach, EMS, Zürich.Google Scholar
[3]Barbour, A. D. and Granovsky, B. L. (2005) Random combinatorial structures: The convergent case J. Combin. Theory Ser. A 109 203–220.CrossRefGoogle Scholar
[4]Betz, V. and Ueltschi, D. (2009) Spatial random permutations and infinite cycles Commun. Math. Phys. 285 469–501.Google Scholar
[5]Betz, V. and Ueltschi, D. (2011) Spatial random permutations with small cycle weights Probab. Theory. Rel. Fields 149 191–222.CrossRefGoogle Scholar
[6]Betz, V. and Ueltschi, D. (2011) Spatial random permutations and Poisson–Dirichlet law of cycle lengths Electron. J. Probab. 16 1173–1192.Google Scholar
[7]Betz, V., Ueltschi, D. and Velenik, Y. (2011) Random permutations with cycle weights Ann. Appl. Probab. 21 312–331.Google Scholar
[8]Bogdanas, K. and Manstavičius, E. (2012) Stochastic processes on weakly logarithmic assemblies. In Analytic and Probabilistic Methods in Number Theory 5 (Laurinčikas, A.et al., eds), Kubilius Memorial Volume, TEV, Vilnius, pp. 69–80.Google Scholar
[9]Ercolani, N. M. and Ueltschi, D. (2014) Cycle structure of random permutations with cycle weights Random Struct. Alg. 44 109–133.Google Scholar
[10]Erdős, P. and Turén, P. (1965) On some problems of a statistical group theory I Z. Wahrsch. Verw. Gebiete 4 175–186.Google Scholar
[11]Flajolet, P. and Sedgewick, R. (2008) Analytic Combinatorics, Cambridge University Press.Google Scholar
[12]Hambly, B., Keevash, P., O'Connell, N. and Stark, D. (2000) The characteristic polynomial of a random permutation matrix Stoch. Process. Appl. 90 335–346.CrossRefGoogle Scholar
[13]Hughes, C., Najnudel, J., Nikeghball, A. and Zeindler, D. (2013) Random permutation matrices under the genereralized Ewens measure Ann. Appl. Probab. 23 987–1024.Google Scholar
[14]Kargina, T. (2007) Additive functions on permutations and the Ewens probability Šiauliai Math. Semin. 10 33–41.Google Scholar
[15]Kargina, T. (2009) Asymptotic distributions of the number of restricted cycles in a random per-mutation Lietuvos matem. rink. Proc. LMS 50 420–425.Google Scholar
[16]Kargina, T. and Manstavičius, E. (2012) Multiplicative functions on Z+n and the Ewens Sampling Formula RIMS Kôkyûroku Bessatsu B34 137–151.Google Scholar
[17]Kargina, T. and Manstavičius, E. (2013) The law of large numbers with respect to Ewens probability. Ann. Univ. Sci. Budapest., Sect. Comp. 39 227–238.Google Scholar
[18]Lugo, M. (2009) Profiles of permutations Electron. J. Combin. 16 1–20.Google Scholar
[19]Lugo, M. (2009) The number of cycles of specified normalized length in permutations. arXiv:0909.2909vIGoogle Scholar
[20]Manstavičius, E. (1996) Additive and multiplicative functions on random permutations Lith. Math. J. 36 400–408.Google Scholar
[21]Manstavičius, E. (2002) Mappings on decomposable combinatorial structures: Analytic approach Combin. Probab. Comput. 11 61–78.CrossRefGoogle Scholar
[22]Manstavičius, E. (2002) Functional limit theorem for sequences of mappings on the symmetric group. In Analytic and Probabilistic Methods in Number Theory 3 (Laurinčikas, A.et al., eds), TEV, Vilnius, pp. 175–187.Google Scholar
[23]Manstavičius, E. (2005) The Poisson distribution for the linear statistics on random permutations Lith. Math. J. 45 434–446.Google Scholar
[24]Manstavičius, E. (2005) Discrete limit laws for additive functions on the symmetric group Acta Math. Univ. Ostraviensis 13 47–55.Google Scholar
[25]Manstavičius, E. (2008) Asymptotic value distribution of additive function defined on the symmetric group Ramanujan J. 17 259–280.Google Scholar
[26]Manstavičius, E. (2009) An analytic method in probabilistic combinatorics Osaka J. Math. 46 273–290.Google Scholar
[27]Manstavičius, E. (2011) A limit theorem for additive functions defined on the symmetric group Lith. Math. J. 51 211–237.Google Scholar
[28]Manstavičius, E. (2012) On total variation approximations for random assemblies. In 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms: AofA'12, DMTCS Proc., pp. 97–108.CrossRefGoogle Scholar
[29]Šiaulys, J. (1996) Convergence to the Poisson law II: Unbounded strongly additive functions. Lith. Math. J. 36 393–404.Google Scholar
[30]Šiaulys, J. (1998) Convergence to the Poisson law III: Method of moments. Lith. Math. J. 38 374–390.Google Scholar
[31]Šiaulys, J. (2000) Factorial moments of distributions of additive functions Lith. Math. J. 40 389–508.CrossRefGoogle Scholar
[32]Šiaulys, J. and Stepanauskas, G. (2008) Some limit laws for strongly additive prime indicators Šiauliai Math. Semin. 3 235–246.Google Scholar
[33]Šiaulys, J. and Stepanauskas, G. (2011) Binomial limit law for additive prime indicators Lith. Math. J. 51 562–572.Google Scholar
[34]Wieand, K. L. (2000) Eigenvalue distributions of random permutation matrices Ann. Probab. 28 1563–1587.CrossRefGoogle Scholar
[35]Wieand, K. L. (2003) Permutation matrices, wreath products, and the distribution of eigenvalues J. Theoret. Probab. 16 599–623.Google Scholar
[36]Zacharovas, V. (2002) The convergence rate to the normal law of a certain variable defined on random polynomials Lith. Math. J. 42 88–107.CrossRefGoogle Scholar
[37]Zacharovas, V. (2004) Distribution of the logarithm of the order of a random permutation Lith. Math. J. 44 296–327.Google Scholar
[38]Zacharovas, V. (2011) Voronoi summation formulae and multiplicative functions on permutations Ramanujan J. 24 289–329.CrossRefGoogle Scholar
[39]Zeindler, D. (2010) Permutation matrices and the moments of their characteristic polynomial Electron. J. Probab. 15 1092–1118.CrossRefGoogle Scholar