Asymptotic Expansions for Products of Weibull Random Variables

We derive an asymptotic expansion for the tail function of the product of n(n∈N) independent identically distributed Weibull random variables. The coefficients of the expansion are obtained using a recursive formula arising from the Laplace method. The resulting expansion provides explicit higher-order correction terms that significantly improve the accuracy of tail approximations for large arguments. These results are useful for both theoretical analysis and practical applications involving extreme-value behavior of products of random variables. The main result of the paper shows that multiplying Weibull distributions yields so-called Weibull-type distributions. It also shows that under multiplication, the shape parameter of the Weibull distribution decreases. This implies that the product of Weibull distributions becomes more heavily tailed. The asymptotic formula for the tail function of the product of Weibull distributions involves rather complicated coefficients. To compute these coefficients, we provide MATLAB (version 9.13.0, R2022b) code. The application of the main result is illustrated with two particular examples.