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Moments of Additive Functions on Random Permutations

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Abstract

We obtain upper and lower bounds for the power moments of additive functions on random permutations. The main ideas of proofs have been originated in probabilistic number theory.

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References

  1. Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic Combinatorial Structures. EMS Monograph Series, vol. 1. Zürich (2003)

  2. Babu, G.J., Manstavičius, E.: Processes with independent increments for the Ewens sampling formula. Ann. Inst. Stat. Math. 54(3), 607–620 (2002)

    Article  MATH  Google Scholar 

  3. Elliott, P.D.T.A.: High power analogues of the Turán-Kubilius inequality and an application to number theory. Can. J. Math. 32, 893–907 (1980)

    MATH  Google Scholar 

  4. Hambly, B.M., Keevash, P., O’Connell, N., Stark, D.: The characteristic polynomial of a random permutation matrix. Stoch. Process. Appl. 90, 335–346 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hildebrand, A.: Sur les moments d’une fonction additive. Ann. Inst. Fourier 33(3), 1–22 (1983)

    MATH  MathSciNet  Google Scholar 

  6. Indlekofer, K.-H.: Über Verallgemeinerungen der Turán-Kubilius Ungleichung. Acta Arith. 52, 67–73 (1989)

    MathSciNet  Google Scholar 

  7. Kubilius, J.: Probabilistic Methods in the Theory of Numbers, Transl. Math. Monographs, vol. 11. Am. Math. Soc., Providence (1964)

  8. Kubilius, J.: On the estimate of the second central moment for arbitrary additive arithmetic functions. Liet. Mat. Rink. 23, 110–117 (1983) (Russian)

    MathSciNet  Google Scholar 

  9. Manstavičius, E.: Inequalities for the p-th moment, 0<p<2, of a sum of independent random variables. In: Gečiauskas, E. (ed.) Abstracts of Communications, vol. III, pp. 200–201. Third International Vilnius Conference on Probability Theory and Mathematical Statistics, Vilnius, Lithuania, June 22–27, 1981. Institute of Mathematics and Cybernetics, Vilnius (1981)

    Google Scholar 

  10. Manstavičius, E.: Inequalities for the p-th moment 0<p<2, of a sum of independent random variables. Lith. Math. J. 22(1), 64–67 (1982)

    Article  MATH  Google Scholar 

  11. Manstavičius, E.: Additive and multiplicative functions on random permutations. Lith. Math. J. 36(4), 400–408 (1996)

    Article  MATH  Google Scholar 

  12. Manstavičius, E.: The law of iterated logarithm for random permutations. Lith. Math. J. 38(2), 160–171 (1998)

    Article  MATH  Google Scholar 

  13. Manstavičius, E.: Value concentration of additive functions on random permutations. Acta Appl. Math. 79, 1–8 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Manstavičius, E.: Asymptotic value distribution of additive functions defined on the symmetric group (2005, submitted, 23 p.)

  15. Manstavičius, E.: Conditional probabilities in combinatorics. The cost of dependence. In: Huškova, M., Janžura, M. (eds.) Prague Stochastics 2006. Proceedings of the Joint Session of 7th Prague Symposium on Asymptotic Statistics and 15th Prague Conference on Information Theory, Statistical Decison Functions and Random Processes, August 21–25, 2006, Prague, pp. 523–532. Charles University, Prague (2006)

    Google Scholar 

  16. Rosenthal, H.P.: On the subspaces of L p, (p>2) spanned by sequences of independent random variables. Israel J. Math. 8, 273–303 (1970)

    MATH  MathSciNet  Google Scholar 

  17. Ruzsa, I.Z.: On the variance of additive functions. Studies in Pure Math. Mem. of P. Turán, 577–586 (1983)

  18. Ruzsa, I.Z.: Generalized moments of additive functions. J. Number Theory 18, 27–33 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  19. Von Bahr, B., Esseen, C.G.: Inequalities for the r-th absolute moment of a sum of random variables, 1≤r≤2. Ann. Math. Stat. 36(1), 299–303 (1965)

    MATH  Google Scholar 

  20. Wieand, K.: Eigenvalue distributions of random matrices in the permutation group and compact Lie groups. PhD thesis, Harvard University (1998)

  21. Zacharovas, V.: The convergence rate to the normal law of a certain variable defined on random polynomials. Lith. Math. J. 42(1), 88–107 (2002)

    Article  MATH  Google Scholar 

  22. Yakymiv, A.L.: Veroyatnostnye Prilozheniya Tauberovykh Teorem (Probabilistic Applications of the Tauber Theorems). Fizmatlit, Moscow (2005)

    Google Scholar 

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Manstavičius, E. Moments of Additive Functions on Random Permutations. Acta Appl Math 97, 119–127 (2007). https://doi.org/10.1007/s10440-007-9133-y

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  • DOI: https://doi.org/10.1007/s10440-007-9133-y

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