An analytic method in probabilistic combinatorics
Eugenijus Manstavičius
Source: Osaka J. Math. Volume 46, Number 1
(2009), 273-290.
Abstract
We deal with the value distribution problem for the linear combinations of multiplicities of the cycle lengths of a random permutation. To examine the characteristic functions, we derive asymptotic formulas for ratios of the Taylor coefficients of the relevant generating series. The proposed version of analytic method does not require any analytic continuation of these series outside the convergence disk.
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Permanent link to this document: http://projecteuclid.org/euclid.ojm/1235574048
Mathematical Reviews number (MathSciNet): MR2531150
Zentralblatt MATH identifier: 1159.60301
References
R. Arratia, A.D. Barbour and S. Tavaré: Logarithmic Combinatorial Structures: A Probabilistic Approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003.
Mathematical Reviews (MathSciNet): MR2032426
Zentralblatt MATH: 1040.60001
R. Arratia and S. Tavaré: Limit theorems for combinatorial structures via discrete process approximations, Random Structures Algorithms 3 (1992), 321--345.
Mathematical Reviews (MathSciNet): MR1164844
G.J. Babu and E. Manstavičius: Brownian motion for random permutations, Sankhyā Ser. A 61 (1999), 312--327.
Mathematical Reviews (MathSciNet): MR1743541
G.J. Babu, E. Manstavičius and V. Zacharovas: Limiting processes with dependent increments for measures on symmetric group of permutations; in Probability and Number Theory---Kanazawa 2005, Adv. Stud. Pure Math. 49, Math. Soc. Japan, Tokyo, 2007, 41--67.
Mathematical Reviews (MathSciNet): MR2405597
Zentralblatt MATH: 1145.60007
R. Durrett, B.L. Granovsky and S. Gueron: The equilibrium behavior of reversible coagulation-fragmentation processes, J. Theoret. Probab. 12 (1999), 447--474.
Mathematical Reviews (MathSciNet): MR1684753
Zentralblatt MATH: 0930.60094
Digital Object Identifier: doi:10.1023/A:1021682212351
P.D.T.A. Elliott: Probabilistic Number Theory. I, II, Springer, New York, 1979/80.
Mathematical Reviews (MathSciNet): MR551361
Zentralblatt MATH: 0431.10029
P. Erdős and P. Turán: On some problems of a statistical group-theory. II, Acta Math. Acad. Sci. Hungar. 18 (1967), 151--163.
Mathematical Reviews (MathSciNet): MR207810
Digital Object Identifier: doi:10.1007/BF02020968
W.J. Ewens: The sampling theory of selectively neutral alleles, Theoret. Population Biology 3 (1972), 87--112.
Mathematical Reviews (MathSciNet): MR325177
Digital Object Identifier: doi:10.1016/0040-5809(72)90035-4
P. Flajolet and A. Odlyzko: Singularity analysis of generating functions, SIAM J. Discrete Math. 3 (1990), 216--240.
Mathematical Reviews (MathSciNet): MR1039294
Zentralblatt MATH: 0712.05004
Digital Object Identifier: doi:10.1137/0403019
G.A. Freiman and B.L. Granovsky: Asymptotic formula for a partition function of reversible coagulation-fragmentation processes, Israel J. Math. 130 (2002), 259--279.
Mathematical Reviews (MathSciNet): MR1919380
Zentralblatt MATH: 1003.60009
Digital Object Identifier: doi:10.1007/BF02764079
V.L. Goncharov: On the distribution of cycles in permutations, Dokl. Acad. Nauk SSSR 35 (1942), 299--301, in Russian.
G. Halász: Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365--403.
Mathematical Reviews (MathSciNet): MR230694
Digital Object Identifier: doi:10.1007/BF01894515
A. Hildebrand: On the limit distribution of discrete random variables, Probab. Theory Related Fields 75 (1987), 67--76.
Mathematical Reviews (MathSciNet): MR879552
Digital Object Identifier: doi:10.1007/BF00320081
N.S. Johnson, S. Kotz and N. Balakrishnan: Discrete Multivariate Distributions, Wiley, New York, 1997.
V.F. Kolchin: Random Mappings, Optimization Software, New York, 1986.
Mathematical Reviews (MathSciNet): MR865130
V.P. Kolchin and V.P. Chistyakov: On the cyclic structure of random permutations, Mat. Zametki 18 (1975), 929--938.
Mathematical Reviews (MathSciNet): MR402852
J. Kubilius: Probabilistic Methods in the Theory of Numbers, Translations of Mathematical Monographs 11, Amer. Math. Soc., Providence, R.I., 1964.
Mathematical Reviews (MathSciNet): MR160745
Zentralblatt MATH: 0133.30203
E. Manstavičius: Additive and multiplicative functions on random permutations, Lith. Math. J. 36 (1996), 400--408.
Mathematical Reviews (MathSciNet): MR1456921
E. Manstavičius: The Berry-Esseen bound in the theory of random permutations, Ramanujan J. 2 (1998), 185--199.
Mathematical Reviews (MathSciNet): MR1642877
Digital Object Identifier: doi:10.1023/A:1009722210949
E. Manstavičius: A Tauber theorem and multiplicative functions on permutations; in Number Theory in Progress, Vol. 2 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 1999, 1025--1038.
Mathematical Reviews (MathSciNet): MR1689557
E. Manstavičius: Mappings on decomposable combinatorial structures: analytic approach, Combin. Probab. Comput. 11 (2002), 61--78.
Mathematical Reviews (MathSciNet): MR1888183
Zentralblatt MATH: 1001.60005
E. Manstavičius: Value concentration of additive functions on random permutations, Acta Appl. Math. 79 (2003), 1--8.
Mathematical Reviews (MathSciNet): MR2021871
Digital Object Identifier: doi:10.1023/A:1025812604540
E. Manstavičius: Asymptotic value distribution of additive function defined on the symmetric group, Ramanujan J. 17 (2008), 259--280.
Mathematical Reviews (MathSciNet): MR2452653
Digital Object Identifier: doi:10.1007/s11139-007-9111-z
V.V. Petrov: Sums of Independent Random Variables, Springer, New York, 1975.
Mathematical Reviews (MathSciNet): MR388499
V. Zacharovas: Voronoi summation formulae and multiplicative functions on permutations, (2004), submitted.
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Osaka Journal of Mathematics
