A comparative study of sampling techniques for generating correlated multivariate normal random variates

This study presents a comparative evaluation of four algorithms for generating correlated multivariate normal random variates. The selected univariate standard normal generation techniques include the Central Limit Theorem approach, the Box–Muller transformation, the Simple Inverse Transform Approximation, and the Polynomial Ratio Approximation. Each algorithm was implemented in Python and validated using uniformly distributed pseudo-random inputs. Correlated multivariate normal data were produced by embedding the uncorrelated samples through the Cholesky decomposition technique. The generated variates were statistically verified using chi-square goodness-of-fit tests, correlation analyses, and coefficient of determination (R square) evaluations. Results indicate that the SITA achieved the highest accuracy and stability across both univariate and multivariate tests, with R square 0.998 and 0.996, followed closely by the Box–Muller transformation with R square 0.993. In contrast, the CLT based, and PRA methods exhibited reduced accuracy, with R square 0.982 and 0.941, respectively, particularly in higher-order variates. The Cholesky decomposition effectively preserved the target correlation structure with minimal deviation (maximum deviation is less than 0.02) between theoretical and empirical coefficients. These findings demonstrate that the combination of Simple Inverse Transform Approximation with Cholesky embedding provides a computationally efficient and statistically reliable framework for generating correlated multivariate normal samples applicable to Monte Carlo simulations, reliability engineering, and stochastic modeling.