RENEWAL, MODULATION AND CRUCIAL EVENTS

Many experimental time series exhibit a stochastic behavior that can be represented by a two-state process. Examples include the gating currents of single ionic channels and the intermittent fluorescence of blinking nano-crystals. The interesting property of these processes is that the waiting time distributions in the two states are distinctly not exponential and rather correspond to inverse power laws.Several proposals have appeared in the literature to account for these non-exponential distributions. We focus our attention on two major ones, called emph{renewal} and emph{modulation}. According to the first proposal, each waiting time interval is generated independently, without any correlation with the others. The second proposal is a form of slow modulation of Poisson processes.This thesis aims at discussing theoretically and numerically the properties of time series generated according to these two approaches to complexity. After illustrating in detail both proposals, we show that, although different, they may lead to identical statistical results, as far as waiting distribution and correlation function are concerned.In this thesis we propose a method of time series analysis, based on aging, which makes it possible to establish whether a given distribution of waiting times obeys modulation or renewal schemes.In the second part of the thesis, we discuss both slow modulation and renewal as sources of diffusion generating fluctuations. The aim here is the detection of the scaling parameters of the diffusion processes under exam; this delicate issue is settled in the framework of the Continuous Time Random Walk which, in the case of slow modulation, requires a suitable generalization. We prove the unexpected result that the diffusion generated by modulation, after a dynamical transition, reaches an asymptotic time regime with the same scaling properties as the renewal model. In fact we find that, even with slow modulation, the departure from ordinary statistical physics is determined by %renewal process:crucial events of an underlying renewal process: these events are responsible for the anomalous scaling and seem to be an unavoidable consequence of any practical way we might adopt to realize modulation. On the basis of these results, we make the conjecture that any experimental time series is characterized by renewal crucial events embedded in a cloud of irrelevant events which exert a camouflage action. This sets a challenge for the identification of the crucial events hidden in experimental time series.