On invariant measures of discrete time filters in the correlated signal-noise case

• 成果类型: Article
• 署名作者: Budhiraja, A
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2002
• 页码: 1096-1113
• 关键词: Ergodicity; STABILITY

摘要

The classical results on the ergodic properties of the nonlinear filter previously have been proved under the crucial assumption that the signal process and the observation noise are independent. This assumption is quite restrictive and many important problems in engineering and stochastic control correspond to filtering models with correlated signal and noise. Unlike the case of independent signal and noise, the filter process in the general correlated case may not be Markov even if the signal is a Markov process. In this work a broad class of discrete time filtering problems with signal-noise correlation is studied. It is shown that the pair process (Y-j, pi(j))(jis an element ofN0) is a Feller-Markov process, where (Y-j)(jis an element ofN0) is the observation process and pi(j) is the filter, that is, the conditional distribution of the signal: X-j given past and current observations. It is shown that if the signal process (X-j) has an invariant measure, then so does (Y-j, pi(j)). Finally, it is proved that if (X-j) has a unique invariant measure and the stationary flow corresponding to the signal process is purely nondeterministic, then the pair (Y-j, pi(j)) has a unique invariant measure.