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The Random Walk Metropolis (RWM) algorithm is a Metropolis- Hastings MCMCalgorithm designed to sample from a given target distribution \pi with Lebesguedensity on R^N. RWM constructs a Markov chain by randomly proposing a newposition (the "proposal move"), which is then accepted or rejected according toa rule which makes the chain reversible with respect to \pi. When the dimensionN is large a key question is to determine the optimal scaling with N of theproposal variance: if the proposal variance is too large, the algorithm willreject the proposed moves too often; if it is too small, the algorithm willexplore the state space too slowly. Determining the optimal scaling of theproposal variance gives a measure of the cost of the algorithm as well. Oneapproach to tackle this issue, which we adopt here, is to derive diffusionlimits for the algorithm. Such an approach has been proposed in the seminalpapers [RGG97, RR98]; in particular in [RGG97] the authors derive a diffusionlimit for the RWM algorithm under the two following assumptions: i) thealgorithm is started in stationarity; ii) the target measure $\pi$ is inproduct form. The present paper considers the situation of practical interestin which both assumptions i) and ii) are removed. That is a) we study the case(which occurs in practice) in which the algorithm is started out ofstationarity and b) we consider target measures which are in non-product form.The target measures that we consider arise in Bayesian nonparametric statisticsand in the study of conditioned diffusions. We prove that, out of stationarity,the optimal scaling for the proposal variance is O(N), as it is instationarity. Notice that the optimal scaling in and out of stationatity neednot be the same in general, and indeed they differ e.g. in the case of the MALAalgorithm [KOS16].

Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity