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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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