Self-similarity and Lamperti convergence for families of stochastic processes

Abstract

We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to a number of important families of processes that are not self-similar in the conventional sense. This includes a new class of fractional Hougaard motions defined as moving averages of Hougaard Lévy process, as well as some well-known families of Hougaard Lévy processes such as the Poisson processes, Brownian motions with drift, and the inverse Gaussian processes. Such families have many properties in common with ordinary self-similar processes, including the form of their covariance functions, and the fact that they appear as limits in a Lamperti-type limit theorem for families of stochastic processes.

Fluctuation Scaling and 1/f Noise

“…A double power law has been proposed to account for the scaling properties of these two approaches [19]. We define the mean of the number of individuals per unit area  of habitat and the mean number of individuals per sampling bins of size t,…”

“…with a being a proportionality constant and β a dimensional parameter related to fractal dimension [19]. This equation distinguishes the scaling behaviour of the mean number of individuals per unit area from the mean number per sampling bin.…”