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The correlation integral and the independence of stochastic processes

A generalization of the correlation integral of Grassberger and Procaccia is used to develop a statistic that has the property that it is asymptotically zero if and only if the underlying Gaussian process is independent. The same implication also holds for certain related processes. It is shown that the stastistic is asymptotically normal for weakly dependent stationary processes. An example is given of a zero autocovariance process. It also applies to ARCH (autoregressive conditionally heteroskedastic) random variables.