On the absolute constant in the remainder term estimate in the central limit theorem for Poisson random sums. (English) Zbl 0933.60005

Korolev, V. Yu.; Shorgin, S. Ya.

Kolchin, V. F. (ed.) et al., Probabilistic methods in discrete mathematics. Proceedings of the fourth international Petrozavodsk conference, Petrozavodsk, Russia, June 3–7, 1996. Utrecht: VSP. 305-308 (1997).

An analog of the Berry-Esseen inequality is proved for sums of a random number of independent identically distributed random variables with finite third moment. The number of summands is assumed to be a Poisson random variable independent of the summands. It is demonstrated that the uniform distance between the distribution function of these sums and the normal distribution function with the same mean and variance is not greater than the Lyapunov fraction with central moments replaced by original ones multiplied by a positive absolute constant \(C\leq C_0\) where \(C_0\) is the corresponding constant in the classical Berry-Esseen inequality, \(C_0< 0.7655\). A similar non-uniform estimate is also proved.
For the entire collection see [Zbl 0895.00041].

Reviewer: V.M.Kruglov (Moskva)

MSC:

60F05	Central limit and other weak theorems
60G50	Sums of independent random variables; random walks

Keywords:

Berry-Esseen inequality; uniform distance; Lyapunov fraction