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We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L 2-norm estimates between the Wiener sausage and the Brownian intersection local times.

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Summary Page’s CUSUM test for detecting change in a sequence of independent observations is extended to the general parametric model involving nuisance parameters. The test statistic is the standardized efficient score vector. The model of nested random effects is analyzed in detail.

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