The joint presence of heavy-tailed distribution and long memory in time series always leads to certain trouble in correctly obtaining the statistical characteristics for time series modeling. These two properties i.e., heavy-tailed distribution and long memory, cannot be neglected in time series analysis, because the tail thickness of the distribution and long memory property of the time series are critical in characterizing the essence of the resulting natural or man-made phenomenon of the time series. Meanwhile, the fluctuation of the varying local long memory parameter may be used to capture the internal changes which underlie the externally observed phenomenon. Therefore, in this paper, we proposed to use the variance trend, heavy-tailed distribution, long memory, and local long memory characteristics to analyze a time series recorded as in [1] from tracking the jumps of individual molecules on cell membranes. The tracked molecules are Class I major histocompatibility complex (MHCI) expressed on rat hepatoma cells. The analysis results show that the jump time series of molecular motion on the cell membrane obviously has both heavy-tailed distribution and local long memory characteristics. The tail heaviness parameters, long memory parameters, and the local long memory parameters of ten MHCI molecular jump time series are all summarized with tables and figures in the paper. These reported tables and figures are not only interesting but also important in terms of additional novel insights and characterization of the time series under investigation.
Time series forecasting: problem of heavy-tailed distributed noise