  Among Monte Carlo techniques, the importance sampling requires fine tuning of
a proposal distribution, which is now fluently resolved through iterative
schemes. The Adaptive Multiple Importance Sampling (AMIS) of Cornuet et al.
(2012) provides a significant improvement in stability and effective sample
size due to the introduction of a recycling procedure. However, the consistency
of the AMIS estimator remains largely open. In this work we prove the
convergence of the AMIS, at a cost of a slight modification in the learning
process. Contrary to Douc et al. (2007a), results are obtained here in the
asymptotic regime where the number of iterations is going to infinity while the
number of drawings per iteration is a fixed, but growing sequence of integers.
Hence some of the results shed new light on adaptive population Monte Carlo
algorithms in that last regime.
