ABOUT THE APPLICATION OF THE STEIN–TIKHOMIROV METHOD IN THE THEORY OF BRANCHING RANDOM PROCESSES

Galton-Watson branching random processes have many practical applications. When studying the structural and asymptotic structure of these processes, generating functions, characteristic functions, and Laplace substitutions from mathematical apparatus are widely used. In this paper we show the application of a certain method proposed by Charles Stein for proving limit theorems. Studying the speed of convergence in the central limit theorem for stationary quantities satisfying Rosenblat's mixed condition, C. Stein used a certain differential identity for the difference between the corresponding distribution functions. Later, this method of his was modified by A. Tikhomirov in terms of characteristic functions. Currently, this method is called the Stein-Tikhomirov (S-T) method and is widely used in the field of limit theorems.高尔顿-沃森分支随机过程（Galton-Watson branching random processes）拥有诸多实际应用场景。在研究该类过程的结构与渐近结构时，数学分析工具中的生成函数、特征函数及拉普拉斯代换得到了广泛应用。本文阐述了查尔斯·斯坦（Charles Stein）提出的某一方法在极限定理证明中的应用。在针对满足罗森布拉特（Rosenblat）混合条件的平稳量研究中心极限定理的收敛速度时，C. 斯坦借助针对对应分布函数之差的某一微分恒等式开展了相关分析。后续，A. 季霍米罗夫（A. Tikhomirov）基于特征函数对该方法进行了改进。目前，该方法被称为斯坦-季霍米罗夫（Stein-Tikhomirov，S-T）方法，并在极限定理研究领域得到了广泛应用。