On the accuracy of the normal approximation to distributions of Poisson random sums

Two-sided bounds were constructed for the constant in the Berry–Esseen inequality for Poisson random sums of independent identically distributed random variables with finite absolute moments of order $2+\delta$ with $\delta\in(0,1]$. The lower bounds were obtained for the first time. For the case $0<\delta<1$, the upper bounds were sharpened, and the nonuniform estimates were proved.