Random Walks Problems

Average Shape of the Excursion

Investigation of a wide range of complex systems involve the measure of some scalar observable over long time intervals, during which the signal exhibits nontrivial fluctuations around some average value or avalanche-like bursts of activity separated by quiescent intervals. The statistical features of such fluctuations reflect the properties of the dynamics that generates them, and represent a key point for understanding the system under investigation. We focused on the average shape of a fluctuation (that, for a time series x(t), is the average value of x at time t between two successive returns to a reference value, separated by a time interval T), showing that it contains crucial pieces of information about the nature of the underlying process. In terms of the stochastic process this quantity is the average excursion of a trajectory. For large classes of stochastic processes we find that the average excursion scales as a power of the interval T, time a scaling function f(t/T). The scaling function f(s) is to a large extent independent of the details of the single increment distribution, while it encodes relevant statistical information on the presence and nature of temporal correlations in the process.

An example: Random Walk

Average shape of fluctuations


  • Average Shape of a Fluctuation: Universality in Excursions of Stochastic Processes Andrea Baldassarri, Francesca Colaiori, and Claudio Castellano, Phys. Rev. Lett. 90, 060601 (2003)
  • Average trajectory of returning walks Francesca Colaiori, Andrea Baldassarri, and Claudio Castellano, Phys. Rev. E 69, 041105 (2004)