Distributional Properties and Higher-Order Moments of the Product of Multiple Independent Normal Random Variables

Abstract

    This study investigates the distributional characteristics and higher-order moments of the product of multiple independent and identically distributed (i.i.d.) standard normal random variables. While the distribution of the product of two normal variables is classical, generalizing to n-fold products presents analytical and computational challenges. We derive exact expressions for higher-order moments—such as variance, skewness, and kurtosis—and demonstrate their exponential growth with dimension. A key mathematical contribution lies in the transformation of the multiplicative process via the logarithmic function, allowing the application of the Central Limit Theorem and approximation of the product as a signed log-normal variable. We also apply the Large Deviation Principle to describe rare-event probabilities of the log-product. These theoretical results are supported by a robust simulation framework that empirically validates the asymptotic behavior, highlighting how the product distribution becomes increasingly heavy-tailed and sharply concentrated near zero as n increases. Our findings offer insights into multiplicative processes and provide practical tools for applications in finance, signal processing, and neural networks. Future directions include generalization to correlated and non-Gaussian variables, and advanced tail approximations.