Geodesic
Limit behavior of the order statistics on the cycle lengths of random $A$-permutations
Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 148-160 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a random permutation $\tau_n$ uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths lie in a fixed set $A$ (the so-called $A$-permutations). Let $\zeta_n$ be the total number of cycles, and let $\eta_n(1)\leq\eta_n(2)\leq\dots\leq\eta_n(\zeta_n)$ be the ordered sample of cycle lengths of the permutation $\tau_n$. We consider a class of sets $A$ with positive density in the set of natural numbers. We study the asymptotic behavior of $\eta_n(m)$ with numbers $m$ in the left-hand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence $\eta_n(m)$ dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340–357] who considered the case $A=\mathbf N$.
Keywords: random $A$-permutation, ordered sample for cycle length of a permutation, order statistics. 
@article{TVP_2024_69_1_a7,
     author = {A. L. Yakymiv},
     title = {Limit behavior of the order statistics on the cycle lengths of random $A$-permutations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {148--160},
     year = {2024},
     volume = {69},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a7/}
}
TY  - JOUR
AU  - A. L. Yakymiv
TI  - Limit behavior of the order statistics on the cycle lengths of random $A$-permutations
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2024
SP  - 148
EP  - 160
VL  - 69
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a7/
LA  - ru
ID  - TVP_2024_69_1_a7
ER  - 
%0 Journal Article
%A A. L. Yakymiv
%T Limit behavior of the order statistics on the cycle lengths of random $A$-permutations
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2024
%P 148-160
%V 69
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a7/
%G ru
%F TVP_2024_69_1_a7
A. L. Yakymiv. Limit behavior of the order statistics on the cycle lengths of random $A$-permutations. Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 148-160. http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a7/

[1] R. Arratia, A. D. Barbour, S. Tavaré, Logarithmic combinatorial structures: a probabilistic approach, EMS Monogr. Math., Eur. Math. Soc. (EMS), Zürich, 2003, xii+363 pp. | DOI | MR | Zbl

[2] R. Arratia, L. Goldstein, L. Gordon, “Poisson approximation and the Chen–Stein method”, With comments and a rejoinder by the authors, Statist. Sci., 5:4 (1990), 403–434 | DOI | MR | Zbl

[3] Ju. V. Bolotnikov, V. N. Sačkov, V. E. Tarakanov, “Asymptotic normality of some variables connected with the cyclic structure of random permutations”, Math. USSR-Sb., 28:1 (1976), 107–117 | DOI | MR | Zbl

[4] L. N. Bol'shev, “Transformations of random variables”, Math. Notes, 1:1 (1967), 72–76 | DOI

[5] B. I. Devyatov, “Limits of admissibility of normal approximations for the Poisson distribution”, Theory Probab. Appl., 14:1 (1969), 170–173 | DOI | MR | Zbl

[6] W. J. Ewens, “The sampling theory of selectively neutral alleles”, Theoret. Population Biol., 3 (1972), 87–112 | DOI | MR | Zbl

[7] V. Gončarov, “On the field of combinatory analysis”, Amer. Math. Soc. Transl. Ser. 2, 19, Amer. Math. Soc., Providence, RI, 1962, 1–46 | DOI | MR | MR | Zbl | Zbl

[8] G. I. Ivchenko, Yu. I. Medvedev, “Random combinatorial objects”, Dokl. Math., 69:3 (2004), 344–347 | MR | Zbl

[9] G. I. Ivchenko, Yu. I. Medvedev, “Random permutations: the general parametric model”, Discrete Math. Appl., 16:5 (2006), 471–478 | DOI | DOI | MR | Zbl

[10] G. I. Ivchenko, M. V. Soboleva, “Some nonequiprobable models of random permutations”, Discrete Math. Appl., 21:4 (2011), 397–406 | DOI | DOI | MR | Zbl

[11] V. F. Kolchin, Random mappings, Transl. Ser. Math. Engrg., Optimization Software, Inc., Publications Division, New York, 1986, xiv+207 pp. | MR | MR | Zbl | Zbl

[12] E. Manstavičius, “Total variation approximation for random assemblies and a functional limit theorem”, Monatsh. Math., 161:3 (2010), 313–334 | DOI | MR | Zbl

[13] E. Manstavičius, “On total variation approximations for random assemblies”, 23rd international meeting on probabilistic, combinatorial, and asymptotic methods for the analysis of algorithms (AofA{'}12) (Montreal, 2012), Discrete Math. Theor. Comput. Sci. Proc., AQ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012, 97–108 | DOI | MR | Zbl

[14] A. G. Postnikov, Introduction to analytic number theory, Transl. Math. Monogr., 68, Amer. Math. Soc., Providence, RI, 1988, vi+320 pp. | DOI | MR | MR | Zbl | Zbl

[15] V. N. Sachkov, “Mappings of a finite set with limitations on contours and height”, Theory Probab. Appl., 17:4 (1973), 640–656 | DOI | MR | Zbl

[16] V. N. Sachkov, Probabilistic methods in combinatorial analysis, Encyclopedia Math. Appl., 56, Cambridge Univ. Press, Cambridge, 1997, x+246 pp. | DOI | MR | MR | Zbl | Zbl

[17] V. N. Sachkov, Combinatorial methods in discrete mathematics, Encyclopedia Math. Appl., 55, Cambridge Univ. Press, Cambridge, 1996, xiv+306 pp. | DOI | MR | MR | Zbl | Zbl

[18] L. A. Shepp, S. P. Lloyd, “Ordered cycle lengths in a random permutation”, Trans. Amer. Math. Soc., 121:2 (1966), 340–357 | DOI | MR | Zbl

[19] A. N. Timashev, Sluchainye komponenty v obobschennoi skheme razmescheniya, Akademiya, M., 2017, 119 pp.

[20] A. N. Timashev, “Random mappings with component sizes from a given set”, Theory Probab. Appl., 64:3 (2019), 481–489 | DOI | DOI | MR | Zbl

[21] A. L. Yakymiv, “On the distribution of the $m$th maximal cycle lengths of random $A$-permutations”, Discrete Math. Appl., 15:5 (2005), 527–546 | DOI | DOI | MR | Zbl

[22] A. L. Yakimiv, Probabilistic applications of Tauberian theorems, Mod. Probab. Stat., VSP, Leiden, 2005, viii+225 pp. | DOI | MR | Zbl | Zbl

[23] A. L. Yakymiv, “A limit theorem for the logarithm of the order of a random $A$-permutation”, Discrete Math. Appl., 20:3 (2010), 247–275 | DOI | DOI | MR | Zbl

[24] A. L. Yakymiv, “On the order of random permutation with cycle weights”, Theory Probab. Appl., 63:2 (2018), 209–226 | DOI | DOI | MR | Zbl