  We consider the optimal scaling problem for high-dimensional random walk
Metropolis (RWM) algorithms where the target distribution has a discontinuous
probability density function. Almost all previous analysis has focused upon
continuous target densities. The main result is a weak convergence result as
the dimensionality d of the target densities converges to infinity. In
particular, when the proposal variance is scaled by $d^{-2}$, the sequence of
stochastic processes formed by the first component of each Markov chain
converges to an appropriate Langevin diffusion process. Therefore optimizing
the efficiency of the RWM algorithm is equivalent to maximizing the speed of
the limiting diffusion. This leads to an asymptotic optimal acceptance rate of
$e^{-2}$ (=0.1353) under quite general conditions. The results have major
practical implications for the implementation of RWM algorithms by highlighting
the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs
algorithms.
