8,690 research outputs found
A brief history of long memory: Hurst, Mandelbrot and the road to ARFIMA
Long memory plays an important role in many fields by determining the
behaviour and predictability of systems; for instance, climate, hydrology,
finance, networks and DNA sequencing. In particular, it is important to test if
a process is exhibiting long memory since that impacts the accuracy and
confidence with which one may predict future events on the basis of a small
amount of historical data. A major force in the development and study of long
memory was the late Benoit B. Mandelbrot. Here we discuss the original
motivation of the development of long memory and Mandelbrot's influence on this
fascinating field. We will also elucidate the sometimes contrasting approaches
to long memory in different scientific communitiesComment: 40 page
Heavy-tailed Distributions In Stochastic Dynamical Models
Heavy-tailed distributions are found throughout many naturally occurring
phenomena. We have reviewed the models of stochastic dynamics that lead to
heavy-tailed distributions (and power law distributions, in particular)
including the multiplicative noise models, the models subjected to the
Degree-Mass-Action principle (the generalized preferential attachment
principle), the intermittent behavior occurring in complex physical systems
near a bifurcation point, queuing systems, and the models of Self-organized
criticality. Heavy-tailed distributions appear in them as the emergent
phenomena sensitive for coupling rules essential for the entire dynamics
Aging renewal theory and application to random walks
The versatility of renewal theory is owed to its abstract formulation.
Renewals can be interpreted as steps of a random walk, switching events in
two-state models, domain crossings of a random motion, etc. We here discuss a
renewal process in which successive events are separated by scale-free waiting
time periods. Among other ubiquitous long time properties, this process
exhibits aging: events counted initially in a time interval [0,t] statistically
strongly differ from those observed at later times [t_a,t_a+t]. In complex,
disordered media, processes with scale-free waiting times play a particularly
prominent role. We set up a unified analytical foundation for such anomalous
dynamics by discussing in detail the distribution of the aging renewal process.
We analyze its half-discrete, half-continuous nature and study its aging time
evolution. These results are readily used to discuss a scale-free anomalous
diffusion process, the continuous time random walk. By this we not only shed
light on the profound origins of its characteristic features, such as weak
ergodicity breaking. Along the way, we also add an extended discussion on aging
effects. In particular, we find that the aging behavior of time and ensemble
averages is conceptually very distinct, but their time scaling is identical at
high ages. Finally, we show how more complex motion models are readily
constructed on the basis of aging renewal dynamics.Comment: 21 pages, 7 figures, RevTe
- …