Relative fixed-width stopping rules for Markov chain Monte Carlo simulations

Markov chain Monte Carlo (MCMC) simulations are commonly employed for estimating features of a target distribution, particularly for Bayesian inference. A fundamental challenge is determining when these simulations should stop. We consider a sequential stopping rule that terminates the simulation when the width of a confidence interval is sufficiently small relative to the size of the target parameter. Specifically, we propose relative magnitude and relative standard deviation stopping rules in the context of MCMC. In each setting, we develop sufficient conditions for asymptotic validity, that is conditions to ensure the simulation will terminate with probability one and the resulting confidence intervals will have the proper coverage probability. Our results are applicable in a wide variety of MCMC estimation settings, such as expectation, quantile, or simultaneous multivariate estimation. Finally, we investigate the finite sample properties through a variety of examples and provide some recommendations to practitioners.Comment: 24 page

Edge States and Quantum Hall Effect in Graphene under a Modulated Magnetic Field

Graphene properties can be manipulated by a periodic potential. Based on the tight-binding model, we study graphene under a one-dimensional (1D) modulated magnetic field which contains both a uniform and a staggered component. New chiral current-carrying edge states are generated at the interfaces where the staggered component changes direction. These edge states lead to an unusual integer quantum Hall effect (QHE) in graphene, which can be observed experimentally by a standard four-terminal Hall measurement. When Zeeman spin splitting is considered, a novel state is predicted where the electron edge currents with opposite polarization propagate in the opposite directions at one sample boundary, whereas propagate in the same directions at the other sample boundary.Comment: 5 pages, 4 figure