Stability with respect to initial conditions in V-norm for nonlinear filters with ergodic observations

We establish conditions for an exponential rate of forgetting of the initial distribution of nonlinear filters in V -norm, allowing for unbounded test functions. The analysis is conducted in an general setup involving nonnegative kernels in a random environment which allows treatment of filters and prediction filters in a single framework. The main result is illustrated on two examples, the first showing that a total variation norm stability result obtained by Douc et al. [4] can be extended to V -norm without any additional assumptions, the second concerning a situation in which forgetting of the initial condition holds in V norm for the filters, but the V -norm of each prediction filter is infinite.