  In several implementations of Sequential Monte Carlo (SMC) methods it is
natural, and important in terms of algorithmic efficiency, to exploit the
information of the history of the samples to optimally tune their subsequent
propagations. In this article we provide a carefully formulated asymptotic
theory for a class of such \emph{adaptive} SMC methods. The theoretical
framework developed here will cover, under assumptions, several commonly used
SMC algorithms. There are only limited results about the theoretical
underpinning of such adaptive methods: we will bridge this gap by providing a
weak law of large numbers (WLLN) and a central limit theorem (CLT) for some of
these algorithms. The latter seems to be the first result of its kind in the
literature and provides a formal justification of algorithms used in many real
data context. We establish that for a general class of adaptive SMC algorithms
the asymptotic variance of the estimators from the adaptive SMC method is
\emph{identical} to a so-called `perfect' SMC algorithm which uses ideal
proposal kernels. Our results are supported by application on a complex
high-dimensional posterior distribution associated with the Navier-Stokes
model, where adapting high-dimensional parameters of the proposal kernels is
critical for the efficiency of the algorithm.
