Using the multivariate long memory (LM) model and Taylor expansions, we find the conditions for convergence of the wavelet correlations between two LM processes on an asymptotic value at low frequencies. These mathematical results, and a least squares estimator of LM parameters, are validated in simulations and applied to neurophysiological (human brain) and financial market time series. Both brain and market systems had multivariate LM properties including a "fractal connectivity" regime of scales over which wavelet correlations were invariantly close to their asymptotic value. This analysis provides efficient and unbiased estimation of long-term correlations in diverse dynamic networks.