  A hidden Markov model is called observable if distinct initial laws give rise
to distinct laws of the observation process. Observability implies stability of
the nonlinear filter when the signal process is tight, but this need not be the
case when the signal process is unstable. This paper introduces a stronger
notion of uniform observability which guarantees stability of the nonlinear
filter in the absence of stability assumptions on the signal. By developing
certain uniform approximation properties of convolution operators, we
subsequently demonstrate that the uniform observability condition is satisfied
for various classes of filtering models with white-noise type observations.
This includes the case of observable linear Gaussian filtering models, so that
standard results on stability of the Kalman--Bucy filter are obtained as a
special case.
