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On the speed of convergence in the central limit theorem of sequential log-likelihood ratio processes. (English) Zbl 0808.62068
Let \(X_ 1, X_ 2,\dots\) be a general dependent sequence of random variables with distribution depending on a parameter vector \(\theta\). Let \(P_{m\theta}\) denote the joint distribution of \(X_ 1, X_ 2,\dots, X_ m\) and \(\Lambda_ m (\theta^*, \theta_ 0)=\log dP_{m\theta^*}/ dP_{m\theta_ 0}\) the log-likelihood ratio statistic, where \(\theta_ 0\) is the true value of \(\theta\) and \(\theta^*= \theta_ m^*\) is a close alternative. Let \((\nu_ n)\) be a sequence of stopping times such that \(\nu_ n/n\) tends to 1 in probability and consider the randomly stopped process \(\Lambda_{\nu_ n} (\theta^*, \theta_ 0)\).
Under a large set of regularity assumptions uniform and nonuniform bounds are derived for the rate of convergence of the suitably normalized sequence \(\overline{\Lambda}_{\nu_ n} (\theta^*, \theta_ 0)\) to the standard normal distribution. A first order autoregressive model and the discrete Ornstein-Uhlenbeck process are studied as particular examples.
MSC:
62L10 Sequential statistical analysis
60F05 Central limit and other weak theorems
60G40 Stopping times; optimal stopping problems; gambling theory
62L12 Sequential estimation
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