A sharp inequality for the variance with respect to the Ewens sampling formula. (English) Zbl 1470.60017
Summary: We examine the variance of a linear statistic defined on the symmetric group endowed with the Ewens probability. Despite the dependence of the summands, it can be bounded from above by a constant multiple of the sum of variances of the summands. We find the exact value of this constant. The analysis of the appearing quadratic forms and eigenvalue search is built upon the exponential matrices and discrete Hahn’s polynomials.
MSC:
| 60C05 | Combinatorial probability |
| 05A16 | Asymptotic enumeration |
| 20P05 | Probabilistic methods in group theory |
Keywords:
random permutation; quadratic form; eigenvalue; discrete Hahn’s polynomials; Turán-Kubilius inequalityReferences:
| [1] | Arratia, R.; Barbour, AD; Tavaré, S., Logarithmic Combinatorial Structures: a Probabilistic Approach (2003), Zürich: EMS Publishing House, Zürich · Zbl 1040.60001 · doi:10.4171/000 |
| [2] | Askey, R., Orthogonal Polynomials and Special Functions (1975), Philadelphia, PA: SIAM, Philadelphia, PA · Zbl 0298.26010 · doi:10.1137/1.9781611970470 |
| [3] | R. Beals and R. Wong, Special Functions, Cambridge Univ. Press, Cambridge, 2012. · Zbl 1222.33001 |
| [4] | Crane, H., The ubiquitous Ewens sampling formula, Stat. Sci., 31, 1-19 (2016) · Zbl 1442.60010 |
| [5] | Ewens, W., The sampling theory of selectively neutral alleles, Theor. Popul. Biol., 3, 87-112 (1972) · Zbl 0245.92009 · doi:10.1016/0040-5809(72)90035-4 |
| [6] | Feng, S., The Poisson-Dirichlet Distribution and Related Topics (2010), Berlin: Springer, Berlin · Zbl 1214.60001 · doi:10.1007/978-3-642-11194-5 |
| [7] | Johnson, NS; Kotz, S.; Balakrishnan, N., Discrete Multivariate Distributions (1997), New York: Wiley, New York · Zbl 0868.62048 |
| [8] | J. Klimavičius and E. Manstavičius, The Turán-Kubilius inequality on permutations, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput., 48:45-51, 2018. · Zbl 1413.60004 |
| [9] | Koekoek, R.; Lesky, PA; Swarttouw, RF, Hypergeometric Orthogonal Polynomials and Their q-Analogues (2010), Berlin: Springer, Berlin · Zbl 1200.33012 · doi:10.1007/978-3-642-05014-5 |
| [10] | Kubilius, J., Estimating the second central moment for strongly additive arithmetic functions, Liet. Mat. Rink., 23, 110-117 (1983) · Zbl 0531.10054 |
| [11] | E. Manstavičius, Conditional probabilities in combinatorics. the cost of dependence, in M. Huškov’a and M. Janžura (Eds.), Prague Stochastics. 7th Prague Symposium on Asymptotic Statistics and 15th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Charles University, Matfyzpress, Prague, 2006, pp. 523-532. |
| [12] | Manstavičius, E., Moments of additive functions defined on the symmetric group, Acta Appl. Math., 97, 119-127 (2007) · Zbl 1117.60011 · doi:10.1007/s10440-007-9133-y |
| [13] | Manstavičius, E., An analytic method in probabilistic combinatorics, Osaka J. Math., 46, 273-290 (2009) · Zbl 1159.60301 |
| [14] | Manstavičius, E., A sharp inequality for the variance with respect to the Ewens sampling formula, Random Struct. Algorithms, 57, 4, 1303-1313 (2020) · Zbl 1456.60034 · doi:10.1002/rsa.20951 |
| [15] | E. Manstavičius and V. Stepanauskas, On variance of an additive function with respect to a generalized Ewens probability, inM. Bousquet-Mélou and M. Soria (Eds.), Proceedings of the 25th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Discrete Math. Theor. Comput. Sci., Proc., BA, Association DMTCS, Nancy, 2014, pp. 301-311. · Zbl 1331.60025 |
| [16] | Manstavičius, E.; Žilinskas, Ž., On a variance related to the Ewens sampling formula, Nonlinear Anal. Model. Control, 16, 4, 453-466 (2011) · Zbl 1271.93141 · doi:10.15388/NA.16.4.14088 |
| [17] | A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Series, Vol. 3: More Special Functions, Gordon and Breach, New York, 1990. · Zbl 0967.00503 |
| [18] | A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Series, Vol. 1, 4th ed., Taylor & Francis, 1998. · Zbl 0511.00044 |
| [19] | Rainville, ED, Special Functions (1960), New York: Macmillan, New York · Zbl 0092.06503 |
| [20] | Watterson, GA, The sampling theory of selectively neutral alleles diffusion model, Adv. Appl. Probab., 6, 463-488 (1974) · Zbl 0289.62020 · doi:10.2307/1426228 |
| [21] | K. Wieand, Eigenvalue Distributions of Random Matrices in the Permutation Group and Compact Lie Groups, PhD thesis, Harvard University, 1998. |
| [22] | Zacharovas, V., Distribution of the logarithm of the order of a random permutation, Liet. Mat. Rink., 44, 372-406 (2004) · Zbl 1128.11040 |
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