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This article is cited in 7 scientific papers (total 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 $k>1+\lambda e$
$$
|B(k)-\Pi_\lambda(k)|\leq\frac{nP_2}2\left(1+\frac{\lambda^2}{(k-1)^2}\right)
\frac{k-1}{k-1-\lambda e}e^{-\lambda}\left(\frac{\lambda e}{k-1}\right)^{k-1}.
$$
Received: 13.04.2004
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
Linking options:
https://www.mathnet.ru/eng/dm160https://doi.org/10.4213/dm160 https://www.mathnet.ru/eng/dm/v16/i2/p148
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