Before revision:  Specifically, we derive the sampling distributions and subsequent hypothesis tests for any measure that can be decomposed into terms that involve independent partial correlations, which we show includes Granger causality and mutual information under a multivariate linear-Gaussian model.
After revision:  Specifically, we derive the sampling distributions and subsequent hypothesis tests for any multivariate correlation-based measures of linear dependence between covariance-stationary time series, including Granger causality and mutual information under a multivariate linear-Gaussian model.