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

TL;DR

This paper proves exponential forgetting of initial conditions in nonlinear filters using V-norm, accommodating unbounded test functions and unifying filters and prediction filters in a general framework.

Contribution

It extends previous stability results to V-norm without extra assumptions and handles cases with infinite prediction filter norms.

Findings

V-norm stability can be achieved without bounded test functions.

Total variation stability results extend to V-norm.

Forgetting of initial conditions occurs even when prediction filter norms are infinite.

Abstract

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.