In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $\zeta$-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous R{e}nyi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e. g., those with the (generalized) negative binomial distribution.

中文翻译：

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混合泊松随机和的大数定律收敛速度的界限

在本文中，构建了混合泊松随机和的大数定律收敛速度的上限。使用 Zolotarev $\zeta$-metric 来衡量极限定律和预极限定律之间的距离。获得的结果将几何随机和的已知收敛率估计（在著名的 R{e}nyi 定理中）扩展到具有混合泊松分布的更广泛的随机指数类别，包括例如具有（广义）负二项式分布的指数。