# Theorems on large deviations in the scheme of allocating identical particles into different cells

Theorems on large deviations in the scheme of allocating identical particles into different cells -- One-dimensional local and integral theorems on large deviations for the number of cells with a fixed number r of particles in the scheme of allocating n identical particles into N different cells, as n, N --» oo in the central domain, are obtained. We consider the scheme of allocating n particles into TV cells, where all allocations of particles into cells have the same probability ( n *^~ 1 ) . This scheme is mentioned in [1] as the Bose-Einstein statistics. Such a kind of allocation is considered in [2] in detail, where some limit theorems on distributions of random variables £r(n, TV), r = 0,1,..., are formulated and proved. Here £ r (n,TV) is the number of cells containing exactly r particles. The classification of possible domains of variation of n, TV and results of investigation of the limit distributions of the random variables £ r (n, TV) in these domains are presented in [3]. In this paper we obtain local and integral theorems on large deviations which give us the asymptotic estimates of the probabilities P{6-(n,TV) = &}, P{£ r (n,TV) > k} and P{£ r (n,TV) < k} in the central domain [4] of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

# Theorems on large deviations in the scheme of allocating identical particles into different cells

Discrete Mathematics and Applications, Volume 3 (6) – Jan 1, 1993
14 pages

/lp/de-gruyter/theorems-on-large-deviations-in-the-scheme-of-allocating-identical-QkXOkJHWVf
Publisher
de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.6.635
Publisher site
See Article on Publisher Site

### Abstract

-- One-dimensional local and integral theorems on large deviations for the number of cells with a fixed number r of particles in the scheme of allocating n identical particles into N different cells, as n, N --» oo in the central domain, are obtained. We consider the scheme of allocating n particles into TV cells, where all allocations of particles into cells have the same probability ( n *^~ 1 ) . This scheme is mentioned in [1] as the Bose-Einstein statistics. Such a kind of allocation is considered in [2] in detail, where some limit theorems on distributions of random variables £r(n, TV), r = 0,1,..., are formulated and proved. Here £ r (n,TV) is the number of cells containing exactly r particles. The classification of possible domains of variation of n, TV and results of investigation of the limit distributions of the random variables £ r (n, TV) in these domains are presented in [3]. In this paper we obtain local and integral theorems on large deviations which give us the asymptotic estimates of the probabilities P{6-(n,TV) = &}, P{£ r (n,TV) > k} and P{£ r (n,TV) < k} in the central domain [4] of

### Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

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