  In this article we study asymptotic properties of weighted samples produced
by the auxiliary particle filter (APF) proposed by pitt and shephard (1999).
Besides establishing a central limit theorem (CLT) for smoothed particle
estimates, we also derive bounds on the Lp error and bias of the same for a
finite particle sample size. By examining the recursive formula for the
asymptotic variance of the CLT we identify first-stage importance weights for
which the increase of asymptotic variance at a single iteration of the
algorithm is minimal. In the light of these findings, we discuss and
demonstrate on several examples how the APF algorithm can be improved.
