This paper deals with the semiparametric estimation of multivariate long-range dependent processes. The parameters of interest in the model are the vector of the long-range dependence parameters and the long-run covariance matrix. The proposed multivariate wavelet-based Whittle estimation is shown to be consistent for the estimation of both the long-range dependence and the covariance matrix. A simulation study confirms the satisfying behaviour of the estimation, which improves the univariate estimation and gives similar results than multivariate Fourier-based procedure. For real data applications, the correlation between time series is an important feature. Usual estimations can be highly biased due to phase-shifts caused by the differences in the properties of autocorrelation in the processes. The long-run covariance matrix provides an interesting estimator for characterizing coupling between time series, also called functional connectivity in neuroscience. A real data application in neuroscience highlights the utility of the wavelets-based method, which is more flexible than Fourier-based procedures. Time series measuring the brain activity are analysed, so as to obtain the characterization of their long-memory behaviour and a measure of the functional connectivity of the brain.