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On the accuracy of the normal approximation to compound Poisson distributions. (English) Zbl 1298.60033
Theory Probab. Appl. 58, No. 1, 138-158 (2014); translation from Teor. Veroyatn. Primen. 58, No. 1, 152-176 (2013).
Let $$X_{1},X_{2},\dots$$ be independent identically distributed random variables with a common distribution function $$F$$. Let $$N_{\lambda}$$ be a random variable with the Poisson distribution with parameter $$\lambda>0$$ . Let $$N_{\lambda}$$ be independent of the sequence $$X_{1},X_{2},\dots$$ for each $$\lambda>0$$. The distribution $$F_{\lambda}$$ of the random variable $$S_{\lambda}=X_{1}+\cdot\cdot\cdot+X_{N_{\lambda}}$$ is called a compound Poisson distribution. The asymptotic normality of compound Poisson distributions means that the uniform distance $\Delta_{\lambda}\left( F\right) =\sup_{x}\left| F_{\lambda}\left( x\right) -\Phi\left( x\right) \right| ,$ where $$\Phi$$ is the standard normal distribution function, tends to zero as $$\lambda$$ grows.
Under some assumptions on $$F$$, the author studies the accuracy of the normal approximation of compound Poisson distributions: $$\Delta_{\lambda}\left( F\right) \rightarrow0$$ as $$\lambda\rightarrow\infty$$.

##### MSC:
 60F05 Central limit and other weak theorems 60F99 Limit theorems in probability theory
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