On the accuracy of approximation in the Poisson limit theorem

D. N. Karymov

Diskretnaya Matematika, Tome 16 (2004) no. 2, pp. 148-159

Résumé

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 $k>1+\lambda e$ $$ |B(k)-\Pi_\lambda(k)|\leq\frac{nP_2}2\left(1+\frac{\lambda^2}{(k-1)^2}\right) \frac{k-1}{k-1-\lambda e}e^{-\lambda}\left(\frac{\lambda e}{k-1}\right)^{k-1}. $$