Convergence in Distribution in Metric Spaces
We write a statistic as a functional on the sample paths of a stochastic process in order to break an analysis of the statistic into two parts: the study of continuity properties of the functional; the study of the stochastic process as a random element of a space of functions. The method has its greatest appeal when many different statistics can be written as functionals on the same process, or when the process has a form that suggests a simple approximation, as in the goodness-of-fit example from Chapter I. There we expressed various statistics as functionals on the empirical process U n , which defines a random element of D[0, 1]. Doob’s heuristic argument suggested that U n should behave like a brownian bridge, in some distributional sense.
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