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Teoriya Veroyatnostei i ee Primeneniya, 1993, Volume 38, Issue 2, Pages 460–470
(Mi tvp3958)
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This article is cited in 5 scientific papers (total in 5 papers)
Short Communications
On large deviations in the Poisson approximation
V. A. Statulyavichus, A. K. Aleshkyavichenea a Institute of Mathematics and Informatics
Abstract:
This paper proves a general lemma comparing the behavior of probabilities of large deviations $\mathbf{P}(X\ge x)$ of a random variable $X$ against the Poisson distribution $1-P(x,\lambda)$ ($\lambda$ is the parameter of the Poisson distribution). When upper bounds are known for the factorial cumulants $\widetilde\Gamma_k (x)$ of $k$th order:
$$
|\widetilde\Gamma _k (X)|\le\frac{k!\lambda}{\Delta^{k-1}}\quad\text{for }\forall k\ge2
$$
for some $\Delta>0$, then large deviations may be compared in the interval $1\le x-\lambda<\delta\lambda\Delta$, $0<\delta<1$. For such $x$
$$
\frac{\mathbf{P}(X\ge x)}{1-P(x,\lambda)}=e^{L(x)}\biggl(1+\theta_1\frac{1+\lambda}{x}+\theta_2\frac{(x-\lambda)^{3/2}}{\Delta}\biggr),
$$
where $L(x)$ is a power series and $|\theta_i|<C(\delta)$, $i=1,2$.
Keywords:
large deviations, Poisson approximation, factorial moments and cumulants, mixed cumulants, higher correlation functions.
Received: 26.01.1993
Citation:
V. A. Statulyavichus, A. K. Aleshkyavichene, “On large deviations in the Poisson approximation”, Teor. Veroyatnost. i Primenen., 38:2 (1993), 460–470; Theory Probab. Appl., 38:2 (1993), 385–393
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https://www.mathnet.ru/eng/tvp3958 https://www.mathnet.ru/eng/tvp/v38/i2/p460
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