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

This paper deals with the performance of the cross- correlation function (CCF) 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 characteristic timescale of the underlying signal. Although various papers recently have been dedicated to this subject, most authors limit their approach to numerical simulations only, resulting in a lack of generality and no explanation of the difficulties associated with the use of the CCF in many practical 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 construction of this function the low- frequency components of the signals contribute more than the high-frequency components. This result explains also the robustness of the CCF to the broadband noise.