This article is cited in 7 papers

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
$$ B(x)=\mathsf P\biggl\{\sum_{k=1}^nI_k\leq x\biggr\}. $$
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 $k\geq2+\lambda$,
$$ |b_k-\pi_k|\leq\frac{nP_2}2\left(1+\frac{\lambda^2}{(k-2)^2}\right) e^{-\lambda}\left(\frac{\lambda e}{k-2}\right)^{k-2}, $$
and for $k>1+\lambda e$
$$ |B(k)-\Pi_\lambda(k)|\leq\frac{nP_2}2\left(1+\frac{\lambda^2}{(k-1)^2}\right) \frac{k-1}{k-1-\lambda e}e^{-\lambda}\left(\frac{\lambda e}{k-1}\right)^{k-1}. $$


UDC: 519.2

Received: 13.04.2004

DOI: 10.4213/dm160

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

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