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On the speed of convergence in the central limit theorem of log- likelihood ratio processes. (English) Zbl 0792.60016

Summary: Let \(\Theta\), the parameter space, be an open subset of \(R^ k\), \(k \geq 1\). For each \(\theta \in \Theta\), let the r.v.’s \(X_ n\), \(n=1,2,\dots\), be defined on the probability space \(({\mathcal X},{\mathcal F},P_ \theta)\) and take values in \((S,{\mathcal S})\) where \(S\) is a Borel subset of a Euclidean space and \({\mathcal S}\) is the \(\sigma\)-field of Borel subsets of \(S\). For \(h \in R^ k\) and a sequence of p.d. normalizing matrices \(\partial_ n=\partial_ n^{k \times k} (\theta_ 0)\) set \(\theta^*_ n=\theta^*=\theta_ 0+\partial_ n h\), where \(\theta_ 0\) is the true value of \(\theta\), such that \(\theta^*\), \(\theta_ 0 \in \Theta\). Let \(\Delta_ n (\theta^*, \theta_ 0)\) be the log-likelihood ratio of the probability measure \(P_{n \theta^*}\) with respect to the probability measure \(P_{n \theta_ 0}\), where \(P_{n \theta}\) is the restriction of \(P_ \theta\) over \({\mathcal F}_ n=\sigma(X_ 1,X_ 2,\dots,X_ n)\). Under a very general dependence setup we obtain a rate of convergence of the normalized log-likelihood ratio statistic to standard normal variable. Two examples are taken into account.

MSC:

60F05 Central limit and other weak theorems
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