Geodesic
On a number of particles in a marked set of cells in a general allocation scheme
Diskretnaya Matematika, Tome 34 (2022) no. 1, pp. 141-152.

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In a generalized allocation scheme of $n$ particles over $N$ cells we consider the random variable $\eta_{n,N}(K)$ which is the number of particles in a given set consisting of $K$ cells. We prove that if $n, K, N\to\infty$, then under some conditions random variables $\eta_{n,N}(K)$ are asymptotically normal, and under another conditions $\eta_{n,N}(K)$ converge in distribution to a Poisson random variable. For the case when $N\to\infty$ and $n$ is a fixed number, we find conditions under which $\eta_{n,N}(K)$ converge in distribution to a binomial random variable with parameters $n$ and $s=\frac{K}{N}$, $0$, multiplied by a integer coefficient. It is shown that if for a generalized allocation scheme of $n$ particles over $N$ cells with random variables having a power series distribution defined by the function $B(\beta)=\ln(1-\beta)$ the conditions $n,N,K\to\infty$, $\frac{K}{N}\to s$, $N=\gamma\ln(n)+o(\ln(n))$, where $0 s1$, $0\gamma\infty$, are satisfied, then distributions of random variables $\frac{\eta_{n,N}(K)}{n}$ converge to a beta-distribution with parameters $s\gamma$ and $(1-s)\gamma$.
Keywords: generalized allocation scheme, Poisson distribution, Gaussian distribution, binomial distribution, hypergeometric distribution, beta-distribution, local limit theorem. 
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A. N. Chuprunov. On a number of particles in a marked set of cells in a general allocation scheme. Diskretnaya Matematika, Tome 34 (2022) no. 1, pp. 141-152. http://geodesic.mathdoc.fr/item/DM_2022_34_1_a9/

[1] Kolchin V. F., “Odin klass predelnykh teorem dlya uslovnykh raspredelenii”, Lit. mat. sb., 8 (1968), 53–63 | Zbl

[2] Kolchin V. F., Sluchainye grafy, FIZMATLIT, Moskva, 2000, 256 pp.

[3] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P., Sluchainye razmescheniya, FIZMATLIT, Moskva, 1976, 224 pp.

[4] Ivchenko G. I., Medvedev Yu. I., Diskretnye raspredeleniya. Veroyatnostnostatisticheskii spravochnik. Odnomernye raspredeleniya, URSS, Moskva, 2015, 2256 pp.

[5] Kolchin A. V., “Predelnye teoremy dlya obobschennoi skhemy razmescheniya”, Diskretnaya matematika, 18:4 (2003), 148–157

[6] Chuprunov A. N., Fazekas I., “Poisson limit theorems for the generalized allocation scheme”, Ann. Univ. Sci. Budapest, Sect. Comp., 49 (2019), 77–96 | MR | Zbl

[7] Trunov A. N., “Predelnye teoremy v zadachakh razmescheniya odinakovykh chastits po razlichnym yacheikam”, Tr. MIAN SSSR, 177 (1988), 157–175 | Zbl

[8] Chuprunov A. N., Fazekas I., “On the number of empty cells in the allocation scheme of indistinguishable particles”, Ann. univ. Mariae Curie-Sklodowska, LXXXIV:1 (2020), 15–29 | DOI | MR

[9] Timashev A. N., Asimptoticheskie razlozheniya v veroyatnostnoi kombinatorike, Nauchnoe izd-vo TVP, Moskva, 2011, 312 pp.