Consistency of orthogonal series density estimators based on grouped observations

In this correspondence we show that nonparametric orthogonal series estimators of probability densities retain the mean integrated square error (MISE) consistency when observations are grouped to the points of a uniform grid (prebinned). This kind of grouping is typical for computer rounding errors and may also be useful in data compression, before calculating estimates, e.g., using the FFT. The main result shows that MISE consistency holds for all square-integrable densities under mild conditions on the grid step size