On the Central Limit Theorem of Random Sum of Independent Random Variables Using Stein’s Method - Huang, Yichen (My University, 2024)
Stein’s method provides a powerful tool for quantifying the distance between a random variable and a normal distribution without relying on the characteristic function as in the traditional method. The traditional characteristic function method is often challenging when dealing with complicated random variables. Stein’s method was first proposed to solve the problem of normal approximation that the characteristic function cannot solve. This method provides a powerful tool to give an upper bound between any random variable and the normal distribution, which can been seen as a speed of convergence to the normal distribution. The core idea of Stein’s method is to construct a differential equation for the target random variable and analyze its solution. In this thesis, I use Stein’s method to quantify the error bound of the standardized random sum of independent random variables with respect to the approximation of the normal distribution to establish the conditions that need to be met for the Central Limit Theorem to hold. Additionally, the error term is bounded using the Wasserstein distance, which demonstrates Stein’s method’s effectiveness in controlling approximation errors by bounding the expectation of the Stein’s Identity. In addition, the results of Charles Stein’s first paper, published in 1972, which describes the Stein method in detail, are also given in the paper. Stein’s paper gives error bound for the sum of dependent variable sequences and the standard normal under certain conditions.
Keywords: Stein's Method; Central Limit Theorem; Random Sums; Probability