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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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