  We analyse the performance of a recursive Monte Carlo method for the Bayesian
estimation of the static parameters of a discrete--time state--space Markov
model. The algorithm employs two layers of particle filters to approximate the
posterior probability distribution of the model parameters. In particular, the
first layer yields an empirical distribution of samples on the parameter space,
while the filters in the second layer are auxiliary devices to approximate the
(analytically intractable) likelihood of the parameters. This approach relates
the this algorithm to the recent sequential Monte Carlo square (SMC$^2$)
method, which provides a {\em non-recursive} solution to the same problem. In
this paper, we investigate the approximation, via the proposed scheme, of
integrals of real bounded functions with respect to the posterior distribution
of the system parameters. Under assumptions related to the compactness of the
parameter support and the stability and continuity of the sequence of posterior
distributions for the state--space model, we prove that the $L_p$ norms of the
approximation errors vanish asymptotically (as the number of Monte Carlo
samples generated by the algorithm increases) and uniformly over time. We also
prove that, under the same assumptions, the proposed scheme can asymptotically
identify the parameter values for a class of models. We conclude the paper with
a numerical example that illustrates the uniform convergence results by
exploring the accuracy and stability of the proposed algorithm operating with
long sequences of observations.
