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Particle island models (Verg\'e et al., 2013) provide a means ofparallelization of sequential Monte Carlo methods, and in this paper we presentnovel convergence results for algorithms of this sort. In particular weestablish a central limit theorem - as the number of islands and the commonsize of the islands tend jointly to infinity - of the double bootstrapalgorithm with possibly adaptive selection on the island level. For thispurpose we introduce a notion of archipelagos of weighted islands and findconditions under which a set of convergence properties are preserved bydifferent operations on such archipelagos. This theory allows arbitrarycompositions of these operations to be straightforwardly analyzed, providing avery flexible framework covering the double bootstrap algorithm as a specialcase. Finally, we establish the long-term numerical stability of the doublebootstrap algorithm by bounding its asymptotic variance under weak and easilychecked assumptions satisfied for a wide range of models with possiblynon-compact state space.

Convergence properties of weighted particle islands with application to the double bootstrap algorithm