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 Diskr. Mat.: Year: Volume: Issue: Page: Find

 Diskr. Mat., 2004, Volume 16, Issue 2, Pages 148–159 (Mi dm160)

On the accuracy of approximation in the Poisson limit theorem

D. N. Karymov

Abstract: In this paper, we find non-uniform bounds in the Poisson theorem. Let $I_1,\ldots,I_n$ be indicators of independent random events. We set $p_k=\mathsf P\{I_k=1\}=1-\mathsf P\{I_k=0\}$, $0\leq p_k\leq1$, $k=1,\ldots,n$. Let
$$B(x)=\mathsf P\{\sum_{k=1}^nI_k\leq x\}.$$
Let $b_k$ be the jump of the distribution function $B(x)$ at the point $k$. We also set
$$P_1=\frac1n\sum_{k=1}^np_k, \qquad P_2=\frac1n\sum_{k=1}^np_k^2.$$
Let
$$\pi_k=\frac{\lambda^k}{k!}e^{-\lambda}, \qquad k=0,1,2,\ldots,$$
be the jumps of the Poisson distribution function with parameter $\lambda\geq0$, and let
$$\Pi_\lambda(x)=\sum_{k\leq x}\pi_k$$
be the corresponding distribution function.
An example of the results obtained in the paper is formulated as follows.
For $\lambda=nP_1$ and $k\geq2+\lambda$,
$$|b_k-\pi_k|\leq\frac{nP_2}2(1+\frac{\lambda^2}{(k-2)^2}) e^{-\lambda}(\frac{\lambda e}{k-2})^{k-2},$$
and for $k>1+\lambda e$
$$|B(k)-\Pi_\lambda(k)|\leq\frac{nP_2}2(1+\frac{\lambda^2}{(k-1)^2}) \frac{k-1}{k-1-\lambda e}e^{-\lambda}(\frac{\lambda e}{k-1})^{k-1}.$$

DOI: https://doi.org/10.4213/dm160

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English version:
Discrete Mathematics and Applications, 2004, 14:3, 317–327

Bibliographic databases:

UDC: 519.2

Citation: D. N. Karymov, “On the accuracy of approximation in the Poisson limit theorem”, Diskr. Mat., 16:2 (2004), 148–159; Discrete Math. Appl., 14:3 (2004), 317–327

Citation in format AMSBIB
\Bibitem{Kar04} \by D.~N.~Karymov \paper On the accuracy of approximation in the Poisson limit theorem \jour Diskr. Mat. \yr 2004 \vol 16 \issue 2 \pages 148--159 \mathnet{http://mi.mathnet.ru/dm160} \crossref{https://doi.org/10.4213/dm160} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2084577} \zmath{https://zbmath.org/?q=an:1122.60027} \transl \jour Discrete Math. Appl. \yr 2004 \vol 14 \issue 3 \pages 317--327 \crossref{https://doi.org/10.1515/1569392031905593} 

• http://mi.mathnet.ru/eng/dm160
• https://doi.org/10.4213/dm160
• http://mi.mathnet.ru/eng/dm/v16/i2/p148

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This publication is cited in the following articles:
1. Cekanavicius V., “Approximation Methods in Probability Theory”, Approximation Methods in Probability Theory, Universitext, Springer International Publishing Ag, 2016, 1–274
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