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

15 December 2015
Mathieu Gerber
N. Whiteley
ArXiv (abs)PDFHTML
Abstract

We establish conditions for an exponential rate of forgetting of the initial distribution of nonlinear filters in VVV-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 VVV-norm without any additional assumptions, the second concerning a situation in which forgetting of the initial condition holds in VVV-norm for the filters, but the VVV-norm of each prediction filter is infinite.

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