On the accuracy of the normal approximation to compound Poisson distributions. (English) Zbl 1298.60033

Shevtsova, I. G.

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 \(\lambda\rightarrow\infty\).

Reviewer: Wiesław Dziubdziela (Miedziana Gora)

MSC:

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

Keywords:

Poisson random sum; central limit theorem; convergence rate estimate; normal approximation; Berry-Esseen inequality; asymptotically exact constant