This paper investigates the exponential convergence of Langevin distributions and their discrete approximations. The authors analyze the convergence properties of the Langevin diffusion, a continuous-time process used to approximate a target distribution π. They find that for one-dimensional distributions with exponential tails of the form π(x) ∝ exp(−γ|x|β), exponential convergence occurs if and only if β ≥ 1. This result is similar to the behavior of symmetric or random walk algorithms. The paper also examines discrete approximations of the Langevin diffusion, such as the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA). While ULA may not converge for light-tailed distributions (β > 2), MALA is shown to converge exponentially fast under certain conditions. However, even MALA may not converge exponentially fast if the diffusion itself does not. The authors also introduce a truncated version of the algorithm, MALTA, which avoids some of the issues encountered with other methods. The paper discusses the convergence properties of the Langevin diffusion and its discrete approximations, showing that the diffusion converges to π exponentially fast under certain conditions. For example, for distributions in the class E(β, γ), exponential convergence occurs if and only if β ≥ 1. For multidimensional exponential families, exponential convergence is also established under similar conditions. The authors also analyze the convergence of the ULA and MALA algorithms, showing that ULA may not converge for light-tailed distributions, while MALA can fail to converge exponentially fast even if the diffusion does. The paper concludes that the choice of discretization and the properties of the target distribution significantly affect the convergence behavior of these algorithms. The results highlight the importance of understanding the tail behavior of the target distribution and the need for careful selection of discretization parameters to ensure efficient convergence.This paper investigates the exponential convergence of Langevin distributions and their discrete approximations. The authors analyze the convergence properties of the Langevin diffusion, a continuous-time process used to approximate a target distribution π. They find that for one-dimensional distributions with exponential tails of the form π(x) ∝ exp(−γ|x|β), exponential convergence occurs if and only if β ≥ 1. This result is similar to the behavior of symmetric or random walk algorithms. The paper also examines discrete approximations of the Langevin diffusion, such as the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA). While ULA may not converge for light-tailed distributions (β > 2), MALA is shown to converge exponentially fast under certain conditions. However, even MALA may not converge exponentially fast if the diffusion itself does not. The authors also introduce a truncated version of the algorithm, MALTA, which avoids some of the issues encountered with other methods. The paper discusses the convergence properties of the Langevin diffusion and its discrete approximations, showing that the diffusion converges to π exponentially fast under certain conditions. For example, for distributions in the class E(β, γ), exponential convergence occurs if and only if β ≥ 1. For multidimensional exponential families, exponential convergence is also established under similar conditions. The authors also analyze the convergence of the ULA and MALA algorithms, showing that ULA may not converge for light-tailed distributions, while MALA can fail to converge exponentially fast even if the diffusion does. The paper concludes that the choice of discretization and the properties of the target distribution significantly affect the convergence behavior of these algorithms. The results highlight the importance of understanding the tail behavior of the target distribution and the need for careful selection of discretization parameters to ensure efficient convergence.