Extremes of Product of Gaussian Random Variables

We investigate the asymptotic behavior of the product of Gaussian random variables, with a focus on tail probabilities. Although sums of normal variables are well understood through classical limit theorems, their products exhibit significantly more complex behavior and have no simple closed-form distributions. We analyze the extremes of such products by deriving precise asymptotic expressions for the tail probabilities as the threshold tends to infinity. The study covers both centered and shifted but independent Gaussian variables, as well as cases with heterogeneous variances and nonzero expectations. Using transformation techniques, geometric arguments in high-dimensional spaces, and asymptotic analysis based on Gaussian measures, we establish general results describing the decay rate of tail probabilities. The main theorems provide explicit asymptotic formulas that depend on the number of variables, their shifts, and the variance structure. Several corollaries present results for important particular cases, including the case of identically distributed random variables. Three individual examples are provided at the end of the paper to illustrate the resulting asymptotic formulas and compare them with Monte Carlo estimates.