On discrete time ergodic filters with wrong initial data

Abstract

For a class of non-uniformly ergodic Markov chains (X n ) satisfying exponential or polynomial beta-mixing, under observations (Y n ) subject to an IID noise with a positive density, it is shown that wrong initial data is forgotten in the mean total variation topology, with a certain exponential or polynomial rate. Mathematics Subject Classification (2000)60G35 · 62M20 · 93E11 · 93E15

On ergodic filters with wrong initial data

“…On considère le problème de la stabilité du filtre optimal par rapport à sa condition initiale pour un système, aussi bien en temps continu (voir (2), (3)) qu'en temps discret (4), (5). Soit b : R d → R d ; h : R d → R , (W, B) t 0 un mouvement Brownien de dimension (d + )-et en temps discret (W n , B n ) -une suite i.i.d.…”

“…We are studying a generic non-uniformly ergodic case. For more detailed presentation and references see [5,6]. if p = 1 then r is assumed to be large enough.…”

Stable Approximation Schemes for Optimal Filters

“…The authors of [3] use martingale convergence results to prove almost sure stability for sequences of integrals f dπ t , where f is a test function of a particular class whose definition involves both the bounded potentials g t and the kernels κ t in the model [3]. Other authors resort to the analysis of the total variation distance between optimal filters obtained from different initial distributions [19,9,15] and relate stability to other properties of the dynamical system, often connected to the ergodicity of the state process [19,9] or its observability and controllability (see [15] for the analysis of the continuous-time optimal filter). A recent analysis that builds upon [9,19] but employs a different metric (which enables the inspection of integrals f dπ t for f unbounded) can be found in [12].…”

“…The main issue with the methods in [3,19,9,15,12] is that stability is related to sets of conditions which are often hard to verify from the standard construction of the filtering operator in terms of the kernels κ t and the potentials g t . In contrast, the authors of [16] provide a set of relatively simple-to-verify sufficient conditions for the stability of Φ t|0 .…”

On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering

“…It is therefore essential to check whether or not the filtering equation "forgets" any erroneous initial distribution. For a thorough discussion on the stability properties of traditional nonlinear filtering problems with a detailed overview of theoretical developments on this subject, we refer to the book [6] and to the more recent article by Kleptsyna and Veretennikov [15]. Besides the fact that significant progress has been made in the recent years in the rigorous derivation of multiple target tracking nonlinear equations (see for instance [4,17,22,27]), up to our knowledge the stability and the robustness properties of these measure-valued models have never been addressed so far in the literature on the subject.…”