  Particle island models (Verg\'e et al., 2013) provide a means of
parallelization of sequential Monte Carlo methods, and in this paper we present
novel convergence results for algorithms of this sort. In particular we
establish a central limit theorem - as the number of islands and the common
size of the islands tend jointly to infinity - of the double bootstrap
algorithm with possibly adaptive selection on the island level. For this
purpose we introduce a notion of archipelagos of weighted islands and find
conditions under which a set of convergence properties are preserved by
different operations on such archipelagos. This theory allows arbitrary
compositions of these operations to be straightforwardly analyzed, providing a
very flexible framework covering the double bootstrap algorithm as a special
case. Finally, we establish the long-term numerical stability of the double
bootstrap algorithm by bounding its asymptotic variance under weak and easily
checked assumptions satisfied for a wide range of models with possibly
non-compact state space.
