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We discuss the relevance and importance of these results in characterising the filter\u2019s behaviour, e.g. it is signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman\u2013Bucy filtering. We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.<\/jats:p>","DOI":"10.1007\/s00498-023-00357-2","type":"journal-article","created":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T13:02:21Z","timestamp":1684501341000},"page":"835-903","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["On the mathematical theory of ensemble (linear-Gaussian) Kalman\u2013Bucy filtering"],"prefix":"10.1007","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8581-8562","authenticated-orcid":false,"given":"Adrian N.","family":"Bishop","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierre","family":"Del Moral","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,5,19]]},"reference":[{"key":"357_CR1","doi-asserted-by":"publisher","first-page":"399","DOI":"10.5194\/angeo-21-399-2003","volume":"21","author":"JI Allen","year":"2003","unstructured":"Allen JI, Eknes M, Evensen G (2003) An ensemble Kalman filter with a complex marine ecosystem model: hindcasting phytoplankton in the Cretan Sea. 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