arXiv:2610.34531v1 Announce Type: new Abstract: We characterize the high-dimensional convergence of the arithmetic mean from the independence Metropolis-Hastings algorithm initialized at its proposal. Convergence is measured in the worst-case expected absolute error over bounded functions. We prove necessary and sufficient conditions for the mean to converge in relation to the importance weight. We precisely describe the logarithmic cutoff profile of the sharp decay to convergence where the relative entropy between the target and proposal determines the required iteration length. These results make connections between the required iterations of independence Metropolis-Hastings and importance sampling. For concentrating Bayesian posteriors, we illustrate how the required iteration time can be estimated providing concrete guidance for simulations. We justify the logarithmic cutoff analysis by showing a standard cutoff phenomenon is absent and prove that popular convergence diagnostics based on simulation output can validate inaccurate simulations.