Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach

We consider the filtering problem for a Markov chain {Xk, Yk}k≥0 with state X and observation Y . The state process {Xk}k≥0 is an homogeneous Markov chain taking value in a measurable set X equipped with a σ-algebra B(X). We let Q be the transition kernel of the chain. The observations {Yk}k≥0 takes values in a measurable set Y (BY is the associated σ-algebra). For i ≤ j, denote Yi:j , (Yi, Yi+1, · · · , Yj). Similar notation will be used for other sequences. We assume furthermore. that for each k ≥ 1 and given Xk, Yk is independent of X1:k−1,Xk+1:∞, Y1:k−1, and Yk+1:∞. We also assume that for each x ∈ X, the conditional law has a density g(x, ·) with respect to some fixed σ-finite measure on the Borel σ-field B(Y). We denote by φξ,n[y0:n] the distribution of the hidden state Xn conditionally on the observations y0:n def = [y0, . . . , yn], which is given by