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Analyticity, Convergence, and Convergence Rate of Recursive MaximumLikelihood Estimation in Hidden Markov Modelsby Vladislav B. Tadic
This paper considers the asymptotic properties of
the recursive maximumlikelihood estimator for hidden Markov
models. The paper is focused on the analytic properties of the
asymptotic loglikelihood and on the pointconvergence and convergence
rate of the recursive maximumlikelihood estimator.
Using the principle of analytic continuation, the analyticity of the
asymptotic loglikelihood is shown for analytically parameterized
hidden Markov models. Relying on this fact and some results
from differential geometry (Lojasiewicz inequality), the almost
sure point convergence of the recursive maximumlikelihood
algorithm is demonstrated, and relatively tight bounds on the
convergence rate are derived. As opposed to the existing result
on the asymptotic behavior of maximumlikelihood estimation
in hidden Markov models, the results of this paper are obtained
without assuming that the loglikelihood function has an isolated
maximum at which the Hessian is strictly negative definite. 