On recursive estimation for hidden Markov models (Q1382498)

Let \(\{X_k\}_{k=1}^{\infty }\) be a stationary Markov chain on \(\{1,\dots , r\}\) with transition probability matrix \(\{a_{i,j}(\phi )\}, \phi \in \Phi \subseteq R^q\). Let the conditional densities of an observable process \(\{Y_k\}\) belong to some family \(\{f(.,\theta ), \theta \in \Theta \}\) where \(\theta =\theta _i(\phi )\) depends on the state \(i\) of \(X_k\) and on the parameter \(\phi \). It is assumed that \(r\) is known. The author considers a recursive estimation for \(\phi \) from an observation of \(\{Y_k\}\). The recursive formula is \(\phi _{n+1}=P_G (\phi _n+\gamma _n h(Y_{n+1};\phi _n))\) where \(h\) is a function, \(P_G\) is the projection onto the closed, bounded, and convex subset \(G\) of \(\Phi \) and \(\gamma _n =\gamma _0 n^{-\alpha }\), \(\alpha \in ({1\over 2},1]\). It is proved that this estimator converges to the set of stationary points of the Kullback--Leibler information. Further it is shown that under some general conditions an averaged recursive estimator is close to optimal.