Limits of the cross-correlation function in the analysis of short time series

This paper deals with the performance of the cross- correlation function (C CF) for the case of time series which contain only a limited number of data . Often, in this case the data span a time interval shorter than the charac teristic timescale of the underlying signal. Although various papers recent ly have been dedicated to this subject, most authors limit their approach t o numerical simulations only, resulting in a lack of generality and no expl anation of the difficulties associated with the use of the CCF in many prac tical situations. We show here, on the basis of theoretical considerations, that the limits of the CCF can be understood from the fact that in the con struction of this function the low- frequency components of the signals con tribute more than the high-frequency components. This result explains also the robustness of the CCF to the broadband noise.