  This work extends Roberts et al. (1997) by considering limits of Random Walk
Metropolis (RWM) applied to block IID target distributions, with corresponding
block-independent proposals. The extension verifies the robustness of the
optimal scaling heuristic, to tune the acceptance rate to $\approx0.234$, for
any choice of proposal shaping. We upgrade the form of weak convergence from a
finite-dimensional subprocess to the infinite dimensional process. We show that
the optimal shaping (in terms of the decay of autocorrelations of linear
functions) is the variance of the target distribution. We show that this choice
coincides with the optimal shaping in terms of spectral gaps in special cases
where they can be computed. Lastly, we provide some negative guarantees,
showing that RWM performance degrades with higher-order dependence. In such
cases, no tuning of RWM will yield performance comparable to an IID target.
