Random \(a\)-permutations in a parametric model. (Russian. English summary) Zbl 07906874
Summary: We consider permutations of the set \(X_n=\{1,2,\ldots,n\}\) such that lengths of all their cycles are elements of the subset \(A\subset \{1,2,\ldots\}\). For a parametric probability measure on the set \(S_n(A)\) properties of the cycle structure of a random permutation are studied for several types of subset \(A\). The problems of testing corresponding statistical hypotheses on the model are analyzed.
MSC:
| 05A05 | Permutations, words, matrices |
| 60C05 | Combinatorial probability |
| 62F03 | Parametric hypothesis testing |
| 94A60 | Cryptography |
References:
| [1] | Goncharov V.L., “Iz oblasti kombinatoriki”, Izv. AN SSSR. Ser. matem., 8:1 (1944), 3-48 · Zbl 0063.01685 |
| [2] | Bolotnikov Yu.V., Sachkov V.N., Tarakanov V.E., “Asimptoticheskaya normalnost nekotorykh velichin, svyazannykh s tsiklovoi strukturoi sluchainykh podstanovok”, Matem. sb., 99:1 (1976), 121-133 · Zbl 0335.60005 |
| [3] | Bolotnikov Yu.V., Sachkov V.N., Tarakanov V.E., “O nekotorykh klassakh sluchainykh velichin na tsiklakh podstanovok”, Matem. sb., 108:1 (1979), 91-104 · Zbl 0446.60015 |
| [4] | Sachkov V.N., Vvedenie v kombinatornye metody diskretnoi matematiki, 2, MTsNMO, M., 2004, 424 pp. |
| [5] | Ivchenko G.I., Medvedev Yu.I., “Metod V. L. Goncharova i ego razvitie v analize razlichnykh modelei sluchainykh podstanovok”, Teoriya veroyatn. i ee primen., 47:3 (2002), 558-566 · doi:10.4213/tvp3695 |
| [6] | Ivchenko G.I., Medvedev Yu.I., “O sluchainykh podstanovkakh”, Trudy po diskret. matem., 5, 2002, 73-92 |
| [7] | Ivchenko G.I., Medvedev Yu.I., “Sluchainye kombinatornye ob”ekty”, Doklady RAN, 396:2 (2004), 151-154 |
| [8] | Ivchenko G.I., Medvedev Yu.I., “Sluchainye podstanovki: obschaya parametricheskaya model”, Diskret. matem., 18:4 (2006), 105-112 · doi:10.4213/dm75 |
| [9] | Ivchenko G.I., Medvedev Yu.I., Vvedenie v matematicheskuyu statistiku, LKI/URSS, M., 2010, 600 pp. |
| [10] | Petrov V.V., Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987, 320 pp. · Zbl 0621.60022 |
| [11] | Ewens W. I., “The sampling theory of selectively neutral alleles”, Theor. Popul. Biol., 3:1 (1972), 87-112 · Zbl 0245.92009 · doi:10.1016/0040-5809(72)90035-4 |
| [12] | Gruder O., “Zur Theorie der Zerlegung von Permutation in Ziklen”, Ark. Mat., 2:5 (1952), 385-414 · Zbl 0051.24801 · doi:10.1007/BF02590995 |
| [13] | Vatutin V.A., Mikhailov V.G., “Predelnye teoremy dlya chisla pustykh yacheek v ravnoveroyatnoi skheme razmescheniya chastits komplektami”, Teoriya veroyatn. i ee primen., 27:4 (1982), 684-692 · Zbl 0517.60008 |
| [14] | Dzhonson N.L., Kots S., Kemp A., Odnomernye diskretnye raspredeleniya, Per. s angl., BINOM. Laboratoriya znanii, M., 2012, 559 pp. |
| [15] | Yakymiv A.L., “Asimptotika momentov chisla tsiklov sluchainoi \(A\)-podstanovki s ostatochnym chlenom”, Diskret. matem., 31:3 (2019), 114-127 · doi:10.4213/dm1582 |
| [16] | Yakymiv A.L., “Dispersiya chisla tsiklov sluchainoi \(A\)-podstanovki”, Diskret. matem., 32:3 (2020), 135-146 · doi:10.4213/dm1616 |
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