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Maximum likelihood estimation in linear infinite dimensional models. (English) Zbl 0815.62057
Summary: Consider a martingale \((c(t), {\mathcal F}_ t )_{t\geq 0}\) with values in the strong dual \({\mathcal D}'\) of a nuclear space \({\mathcal D}\). Let \(c(t)\) satisfy the functional equation \[ c(t)- c(0)= \int_ 0^ t A'(\theta) c(s) ds+ B'W(t) \] in which \((W(t), {\mathcal F}_ t )_{t\geq 0}\) is a \({\mathcal D}'\) valued Gaussian white noise process and \(B':{\mathcal D}'\to {\mathcal D}'\) and \(A' (\theta): {\mathcal D}'\to {\mathcal D}'\), \(\theta\in \Theta \subset (-\infty, \infty)^ K\), are continuous linear operators. It is shown that under suitable assumptions the initial condition \(c(0)\) can be chosen in such a way that \((c(t), {\mathcal F}_ t )_{t\geq 0}\) becomes an ergodic stationary Markov process and the unknown parameter \(\theta\) can be estimated by the maximum likelihood method. The obtained estimator of \(\theta\) is strongly consistent and satisfies a version of the central limit theorem.

MSC:
62M05 Markov processes: estimation; hidden Markov models
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60G44 Martingales with continuous parameter
46N30 Applications of functional analysis in probability theory and statistics
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