Tail-adaptive generation of random numbers from a gamma-order normal distribution using the Ziggurat algorithm with a multivariate extension

What the paper says

The Ziggurat algorithm is a well-established rejection-sampling method designed for the efficient generation of pseudo-random numbers from unimodal distributions, particularly the standard normal. In this work, we extend and adapt the Ziggurat algorithm to enable the tail-adaptive generation of random numbers from the gamma-order generalized normal distribution |a flexible family characterized by a tail-shaping parameter that governs transitions between light, Gaussian, and heavy-tailed regimes. The resulting algorithm retains the computational speed of the original Ziggurat algorithm while supporting both univariate and multivariate implementations. This extension is especially relevant in simulation-intensive contexts, such as Bayesian modeling, quantitative nance, and machine learning. We provide the mathematical foundation, reproducible implementation details, and extensive benchmarking results that validate the method's efficiency and accuracy. A multivariate extension based on radial decomposition is also introduced, demonstrating the feasibility of generating random variables from symmetric multivariate distributions in practice. To illustrate the practical utility of the proposed algorithm, we present a comprehensive Monte Carlo simulation study evaluating performance across various shape and scale con gurations. Additionally, we apply the method to real-world data from biomedical signal processing, highlighting its robustness and adaptability to empirical settings where tail behavior plays a crucial role.