This paper presents a theoretical analysis of sampling-based Model Predictive Control (MPC), focusing on the convergence properties and optimal covariance design for the Model Predictive Path Integral Control (MPPI) algorithm. The authors demonstrate that MPPI achieves linear convergence rates when the optimization problem is quadratic, which applies to time-varying Linear Quadratic Regulator (LQR) systems. They extend this analysis to more general nonlinear systems and propose a novel sampling-based MPC algorithm called CoVariance-Optimal MPC (CoVO-MPC), which optimally schedules the sampling covariance to improve convergence rates. Empirical results show that CoVO-MPC outperforms standard MPPI by 43-54% in both simulations and real-world quadrotor control tasks. The theoretical analysis provides a foundation for designing an efficient and effective MPC algorithm, with implications for a wide range of applications in robotics and control systems. The paper also discusses the limitations of sampling-based MPC and outlines future research directions, including the extension of the theory to receding-horizon settings and the integration of CoVO-MPC with model-based reinforcement learning frameworks.This paper presents a theoretical analysis of sampling-based Model Predictive Control (MPC), focusing on the convergence properties and optimal covariance design for the Model Predictive Path Integral Control (MPPI) algorithm. The authors demonstrate that MPPI achieves linear convergence rates when the optimization problem is quadratic, which applies to time-varying Linear Quadratic Regulator (LQR) systems. They extend this analysis to more general nonlinear systems and propose a novel sampling-based MPC algorithm called CoVariance-Optimal MPC (CoVO-MPC), which optimally schedules the sampling covariance to improve convergence rates. Empirical results show that CoVO-MPC outperforms standard MPPI by 43-54% in both simulations and real-world quadrotor control tasks. The theoretical analysis provides a foundation for designing an efficient and effective MPC algorithm, with implications for a wide range of applications in robotics and control systems. The paper also discusses the limitations of sampling-based MPC and outlines future research directions, including the extension of the theory to receding-horizon settings and the integration of CoVO-MPC with model-based reinforcement learning frameworks.