Asymptotic filtering for finite state Markov chains
Let \(\{X_n^{\varepsilon }\}\) be a discrete time Markov chain with state space \(S=\{1,\dots ,d\}\), initial distribution \(p_0^{\varepsilon }\), and transition probabilities \(\pi _{ij}^{\varepsilon }=\varepsilon \lambda _{ij}\) \((i\neq j)\), \(\pi _{ii}^{\varepsilon }=1-\varepsilon \lambda _{ii}\). It is assumed that the chain is irreducible and aperiodic. Let the observations \(\{y_n^{\varepsilon }\}_{n=1}^{\infty }\), given \(\{X_n^{\varepsilon }\}_{n=1}^{\infty }\), be independent, with \(P\left (y_n^{\varepsilon }\in dx |\{X_i^{\varepsilon }\}_{n=1}^{\infty }\right )=P\left (y_n^{\varepsilon }\in dx|X_n^{\varepsilon }\right )\), \(P\left (y_n^{\varepsilon }\in dx|X_n^{\varepsilon }=i\right )=\mu _i(dx)\). Define \({\mathcal F}_{y,n}= \sigma \{y_i^{\varepsilon }, 1\leq i\leq n\}\). The filtering means to find the best estimator \(\tilde X_n\) for \(X_n^{\varepsilon }\) given \({\mathcal F}_{y,n}\). The authors present asymptotic formulas for the filtering error \(P_e^{\varepsilon , n}= E\left (1_{X_n^{\varepsilon }\neq \tilde X_n}\right )\). Asymptotically optimal filters are also introduced, which do not depend on the transition rates of the chain.
scientific article; zbMATH DE number 1046566