Overcomplete representations such as wavelets and windowed Fourier expansions have become mainstays of modern statistical data analysis. Here we derive expressions for the mean quadratic risk of shrinkage estimators in the context of general finite frames, which include any fullrank linear expansion of vector data in a finite-dimensional setting. We provide several new results and practical estimation procedures that take into account the geometric correlation structure of frame elements.

We derive bounds on the Kolmogorov distance between the dis- tribution of a random functional of a {0, 1}-valued random sequence and the normal distribution. Our approach, which relies on the general framework of stochastic analysis for discrete-time normal martingales, extends existing results obtained for independent Bernoulli (or Rademacher) sequences. In particular, we obtain Kolmogorov distance bounds for the sum of normalized random sequences without any independence assumption.

The accurate characterization of cortical functional connectivity from Magnetoencephalography (MEG) data remains a challenging problem due to the subjective nature of the analysis, which requires several decisions at each step of the analysis pipeline, such as the choice of a source estimation algorithm, a connectivity metric and a cortical parcellation, to name but a few. Recent studies have emphasized the importance of selecting the regularization parameter in minimum norm estimates with caution, as variations in its value can result in significant differences in connectivity estimates.

Magnetoencephalography and electroencephalography (M/EEG) seed-based connectivity analysis typically requires regions of interest (ROI)-based extraction of measures. M/EEG ROI-derived source activity can be treated in different ways. For instance, it is possible to average each ROI's time series prior to calculating connectivity measures. Alternatively one can compute connectivity maps for each element of the ROI, prior to dimensionality reduction to obtain a single map. The impact of these different strategies on connectivity estimation is still unclear.

Automatic estimation of current dipoles from biomagnetic data is still a problematic task. This is due not only to the ill-posedness of the inverse problem but also to two intrinsic difficulties introduced by the dipolar model: the unknown number of sources and the nonlinear relationship between the source locations and the data. Recently, we have developed a new Bayesian approach, particle filtering, based on dynamical tracking of the dipole constellation.

In this paper, we study fluctuations and precise deviations of cumulative INAR time series, both in a non-stationary and in a stationary regime. The theoretical results are based on the recent mod- convergence theory as presented in Féray et al., 2016. We apply our findings to the construction of approximate confidence intervals for model parameters and to quantile calculation in a risk management context.