Abstract

Since the appearance of Robbins’s paper (1948) the central limit theorem for a sum of a random number of independent identically distributed random variables is one of the most fundamental results in probability, and explains the appearance of the normal distribution in a whole host of diverse applications in mathematics, physics, biology and the social sciences. Compound random sums are extensions of classical random sums when the numbers of independent summands in sums are partial sums of independent identically distributed positive integer-valued random variables, assumed independent of summands of sums. The main aim of this paper is to introduce central limit theorems for normalized compound random sums of independent random variables and establish the convergence rates in types of small-o and large-\(\mathcal{O}\) estimates, in term of Trotter-distance. The obtained results in this paper are extensions of several known ones.

中文翻译：

---

独立随机变量的复合随机和的中心极限定理的收敛速度

摘要

由于罗宾斯（Robbins）（1948）论文的出现，随机数的独立均等分布的随机变量之和的中心极限定理是概率中最基本的结果之一，并解释了整体中正态分布在多种多样的情况下的出现。在数学，物理学，生物学和社会科学中的应用。当总和中的独立求和数是独立的，均等分布的正整数值随机变量的部分总和（假定独立于总和）时，复合随机和是经典随机和的扩展。本文的主要目的是为独立随机变量的归一化复合随机和引入中心极限定理，并确定小o和大\（\ mathcal {O} \）类型的收敛速度以Trotter距离估算。本文获得的结果是几个已知结果的扩展。