  We consider the random walk Metropolis algorithm on $\mathbb{R}^n$ with
Gaussian proposals, and when the target probability measure is the $n$-fold
product of a one-dimensional law. In the limit $n\to\infty$, it is well known
(see [Ann. Appl. Probab. 7 (1997) 110-120]) that, when the variance of the
proposal scales inversely proportional to the dimension $n$ whereas time is
accelerated by the factor $n$, a diffusive limit is obtained for each component
of the Markov chain if this chain starts at equilibrium. This paper extends
this result when the initial distribution is not the target probability
measure. Remarking that the interaction between the components of the chain due
to the common acceptance/rejection of the proposed moves is of mean-field type,
we obtain a propagation of chaos result under the same scaling as in the
stationary case. This proves that, in terms of the dimension $n$, the same
scaling holds for the transient phase of the Metropolis-Hastings algorithm as
near stationarity. The diffusive and mean-field limit of each component is a
diffusion process nonlinear in the sense of McKean. This opens the route to new
investigations of the optimal choice for the variance of the proposal
distribution in order to accelerate convergence to equilibrium (see [Optimal
scaling for the transient phase of Metropolis-Hastings algorithms: The longtime
behavior Bernoulli (2014) To appear]).
