  This paper investigates adaptive importance sampling algorithms for which the
policy, the sequence of distributions used to generate the particles, is a
mixture distribution between a flexible kernel density estimate (based on the
previous particles), and a "safe" heavy-tailed density. When the share of
samples generated according to the safe density goes to zero but not too
quickly, two results are established: (i) uniform convergence rates are derived
for the policy toward the target density; (ii) a central limit theorem is
obtained for the resulting integral estimates. The fact that the asymptotic
variance is the same as the variance of an "oracle" procedure with
variance-optimal policy, illustrates the benefits of the approach. In addition,
a subsampling step (among the particles) can be conducted before constructing
the kernel estimate in order to decrease the computational effort without
altering the performance of the method. The practical behavior of the
algorithms is illustrated in a simulation study.
