  The reversible jump algorithm is a useful Markov chain Monte Carlo method
introduced by Green (1995) that allows switches between subspaces of differing
dimensionality, and therefore, model selection. Although this method is now
increasingly used in key areas (e.g. biology and finance), it remains a
challenge to implement it. In this paper, we focus on a simple sampling context
in order to obtain theoretical results that lead to an optimal tuning procedure
for the considered reversible jump algorithm, and consequently, to easy
implementation. The key result is the weak convergence of the sequence of
stochastic processes engendered by the algorithm. It represents the main
contribution of this paper as it is, to our knowledge, the first weak
convergence result for the reversible jump algorithm. The sampler updating the
parameters according to a random walk, this result allows to retrieve the
well-known 0.234 rule for finding the optimal scaling. It also leads to an
answer to the question: "with what probability should a parameter update be
proposed comparatively to a model switch at each iteration?"
