  Several particle algorithms admit a Feynman-Kac representation such that the
potential function may be expressed as a recursive function which depends on
the complete state trajectory. An important example is the mixture Kalman
filter, but other models and algorithms of practical interest fall in this
category. We study the asymptotic stability of such particle algorithms as time
goes to infinity. As a corollary, practical conditions for the stability of the
mixture Kalman filter, and a mixture GARCH filter, are derived. Finally, we
show that our results can also lead to weaker conditions for the stability of
standard particle algorithms, such that the potential function depends on the
last state only.
