  We establish conditions for an exponential rate of forgetting of the initial
distribution of nonlinear filters in $V$-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 $V$-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.
