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

Part of: Inference from stochastic processes Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  04 April 2017

Mathieu Gerber  and
Nick Whiteley
Mathieu Gerber*
Affiliation:
Harvard University
Nick Whiteley*
Affiliation:
University of Bristol
*
* Current address: School of Mathematics, University of Bristol, University Walk, BristolBS8 1TW, UK. Email address: mathieu.gerber@bristol.ac.uk
** Postal address: School of Mathematics, University of Bristol, University Walk, BristolBS8 1TW, UK.
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Abstract

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. (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.

Keywords

Nonlinear filteringhidden Markov modelrandom environmentV-norm

MSC classification

Primary: 60G35: Signal detection and filtering
Secondary: 62M20: Prediction 93E11: Filtering
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 118 - 133
Copyright
Copyright © Applied Probability Trust 2017 

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