Let $\mathscr{P}(E)$ be the space of probability measures on a measurable
space $(E,\mathcal{E})$. In this paper we introduce a class of nonlinear Markov
chain Monte Carlo (MCMC) methods for simulating from a probability measure
$\pi\in\mathscr{P}(E)$. Nonlinear Markov kernels (see [Feynman--Kac Formulae:
Genealogical and Interacting Particle Systems with Applications (2004)
Springer]) $K:\mathscr{P}(E)\times E\rightarrow\mathscr{P}(E)$ can be
constructed to, in some sense, improve over MCMC methods.

Sequential Monte Carlo (SMC) methods are a widely used set of computational
tools for inference in non-linear non-Gaussian state-space models. We propose a
new SMC algorithm to compute the expectation of additive functionals
recursively. Essentially, it is an online or forward-only implementation of a
forward filtering backward smoothing SMC algorithm proposed in Doucet .et .al
(2000).

In practical nonlinear filtering, the assessment of achievable filtering
performance is important. In this paper, we focus on the problem of efficiently
approximate the posterior Cramer-Rao lower bound (CRLB) in a recursive manner.
By using Gaussian assumptions, two types of approximations for calculating the
CRLB are proposed: An exact model using the state estimate as well as a
Taylor-series-expanded model using both of the state estimate and its error
covariance, are derived. Moreover, the difference between the two approximated
CRLBs is also formulated analytically.

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.