  One main limitation of the existing optimal scaling results for
Metropolis--Hastings algorithms is that the assumptions on the target
distribution are unrealistic. In this paper, we consider optimal scaling of
random-walk Metropolis algorithms on general target distributions in high
dimensions arising from practical MCMC models from Bayesian statistics. For
optimal scaling by maximizing expected squared jumping distance (ESJD), we show
the asymptotically optimal acceptance rate $0.234$ can be obtained under
general realistic sufficient conditions on the target distribution. The new
sufficient conditions are easy to be verified and may hold for some general
classes of MCMC models arising from Bayesian statistics applications, which
substantially generalize the product i.i.d. condition required in most existing
literature of optimal scaling. Furthermore, we show one-dimensional diffusion
limits can be obtained under slightly stronger conditions, which still allow
dependent coordinates of the target distribution. We also connect the new
diffusion limit results to complexity bounds of Metropolis algorithms in high
dimensions.
