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Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 1, Pages 62–72 (Mi tvp685)  

A Case of Uniform Local Limit Theorems with Changing Lattice in a Classical Problem with Balls

V. F. Kolchin

Moscow

Abstract: Each of $n$ balls is deposited in a cell selected at random out of $N$ given cells. The successive selections are mutually independent and the probability of any fixed cell to be selected is equal to $1/N$. Let $\mu_r$ be the number of cells that contain exactly $r$ balls, $r=0,1,…,n$. In [1] the speed of convergence of the distributions of $\mu_r$ to limiting distributions as $n$, $N\to\infty$ was studied. It was found that the distributions of $\mu_r$ have similar behaviour and converge for any fixed $r$ to either normal or Poisson distribution. The only exception was the behaviour of $\mu_1$ in the case $n$, $N\to\infty$ and $n/N\to0$. In this paper we consider that exceptional case. The main feature is the transition of the distribution of $n-\mu_1$ from the lattice of all non-negative integers to the lattice of even non-negative integers as ratio $n^2/N^3$ is varying from $\infty$ to 0.

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English version:
Theory of Probability and its Applications, 1967, 12:1, 57–67

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Received: 18.01.1966

Citation: V. F. Kolchin, “A Case of Uniform Local Limit Theorems with Changing Lattice in a Classical Problem with Balls”, Teor. Veroyatnost. i Primenen., 12:1 (1967), 62–72; Theory Probab. Appl., 12:1 (1967), 57–67

Citation in format AMSBIB
\Bibitem{Kol67}
\by V.~F.~Kolchin
\paper A~Case of Uniform Local Limit Theorems with Changing Lattice in a~Classical Problem with Balls
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 1
\pages 62--72
\mathnet{http://mi.mathnet.ru/tvp685}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=219118}
\zmath{https://zbmath.org/?q=an:0164.48401}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 1
\pages 57--67
\crossref{https://doi.org/10.1137/1112006}


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