We establish conditions for an exponential rate of forgetting of the initial distribution of nonlinear filters in -norm, path-wise along almost all observation sequences. In contrast to previous works, our results allow 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. (2009) can be extended to -norm without any additional assumptions, the second concerning a situation in which forgetting of the initial condition holds in -norm for the filters, but the -norm of each prediction filter is infinite.
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