Geodesic
On a Functional of the Number of Nonoverlapping Chains Appearing in the Polynomial Scheme and Its Connection with Entropy
Matematičeskie zametki, Tome 114 (2023) no. 3, pp. 390-403 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Consider $n$ independent chains consisting of $k$ independent polynomial trials with $M$ outcomes. It is assumed that $n, k \to \infty$ and $\ln(n/M^k)=o(k)$. We find the asymptotics of the normalized logarithm of the number of appearing chains and indicate the connection between this functional and the entropy.
Keywords: number of absent chains, number of empty cells, entropy, Shannon–McMillan–Breiman theorem, random allocations. 
@article{MZM_2023_114_3_a5,
     author = {M. P. Savelov},
     title = {On a {Functional} of the {Number} of {Nonoverlapping} {Chains} {Appearing} in the {Polynomial} {Scheme} and {Its} {Connection} with {Entropy}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {390--403},
     year = {2023},
     volume = {114},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_114_3_a5/}
}
TY  - JOUR
AU  - M. P. Savelov
TI  - On a Functional of the Number of Nonoverlapping Chains Appearing in the Polynomial Scheme and Its Connection with Entropy
JO  - Matematičeskie zametki
PY  - 2023
SP  - 390
EP  - 403
VL  - 114
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MZM_2023_114_3_a5/
LA  - ru
ID  - MZM_2023_114_3_a5
ER  - 
%0 Journal Article
%A M. P. Savelov
%T On a Functional of the Number of Nonoverlapping Chains Appearing in the Polynomial Scheme and Its Connection with Entropy
%J Matematičeskie zametki
%D 2023
%P 390-403
%V 114
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2023_114_3_a5/
%G ru
%F MZM_2023_114_3_a5
M. P. Savelov. On a Functional of the Number of Nonoverlapping Chains Appearing in the Polynomial Scheme and Its Connection with Entropy. Matematičeskie zametki, Tome 114 (2023) no. 3, pp. 390-403. http://geodesic.mathdoc.fr/item/MZM_2023_114_3_a5/

[1] V. F. Kolchin, B. A. Sevastyanov, V. P. Chistyakov, Sluchainye razmescheniya, Nauka, M., 1976 | MR

[2] M. I. Tikhomirova, V. P. Chistyakov, “Asimptoticheskaya normalnost chisel nepoyavivshikhsya znachenii $m$-zavisimykh sluchainykh velichin”, Diskret. matem., 26:2 (2014), 143–158 | DOI | MR

[3] M. I. Tikhomirova, V. P. Chistyakov, “Ob asimptotike momentov chisla nepoyavivshikhsya $s$-tsepochek”, Diskret. matem., 9:1 (1997), 12–29 | DOI | MR | Zbl

[4] V. P. Chistyakov, “Ob asimptoticheskoi normalnosti chisla pustykh yacheek v odnoi skheme razmescheniya chastits komplektami”, Matem. vopr. kriptogr., 5:4 (2014), 129–138 | DOI

[5] M. I. Tikhomirova, “Ob asimptoticheskoi normalnosti chisla nepoyavivshikhsya $s$-tsepochek”, Diskret. matem., 4:2 (1992), 122–129 | MR | Zbl

[6] M. I. Tikhomirova, “Predelnye raspredeleniya chisla nepoyavivshikhsya tsepochek odinakovykh iskhodov”, Diskret. matem., 20:3 (2008), 40–46 | DOI | MR | Zbl

[7] R. L. Dobrushin, “Uproschennyi metod eksperimentalnoi otsenki entropii statsionarnoi posledovatelnosti”, Teoriya veroyatn. i ee primen., 3:4 (1958), 462–464

[8] L. F. Kozachenko, N. N. Leonenko, “O statisticheskoi otsenke entropii sluchainogo vektora”, Probl. peredachi inform., 23:2 (1987), 9–16 | MR | Zbl

[9] A. B. Tsybakov, E. C. van der Meulen, “Root-$n$ consistent estimators of entropy for densities with unbounded support”, Scand. J. Statist., 23:1 (1996), 75–83 | MR

[10] P. H. Algoet, T. M. Cover, “A sandwich proof of the Shannon–McMillan–Breiman theorem”, Ann. Probab., 16:2 (1988), 899–909 | MR

[11] A. M. Zubkov, A. A. Serov, “Polnoe dokazatelstvo universalnykh neravenstv dlya funktsii raspredeleniya binomialnogo zakona”, Teoriya veroyatn. i ee primen., 57:3 (2012), 597–602 | DOI