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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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