Point Processes, Long-Range Dependence and Wavelets

Phenomena that exhibit long-range dependence characteristics have been found to exist in many different areas, such as physics, geology, communications and biology. This chapter shows that the Fano factor can be understood through multiresolution analysis and therefore to propose a timescale-based generalization of this tool. It explains how and why this wavelet analysis of long-range dependent point processes provides with a more versatile and powerful estimator of the long-range dependence parameter. The chapter presents the analysis of such a point process made of a spiketrain of discharges recorded from auditory neurons responding to an acoustic stimulus. The exponents of the power-laws, whatever the statistics are the most meaningful parameters characterizing the long-range dependence phenomenon. The Poisson process plays, for point processes, the reference role the white Gaussian noise does for continuous time random processes. The chapter aims to point out some connections between the standard and the wavelet-based Fano factor.