Self-similar process
Self-similar processes are stochastic processes satisfying a mathematically precise version of the self-similarity property. Several related properties have this name, and some are defined here.
A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension.
Because stochastic processes are random variables with a time and a space component, their self-similarity properties are defined in terms of how a scaling in time relates to a scaling in space.
Distributional self-similarity
Definition
A continuous-time stochastic process is called self-similar with parameter if for all, the processes and have the same law.Examples
- The Wiener process is self-similar with.
- The fractional Brownian motion is a generalisation of Brownian motion that preserves self-similarity; it can be self-similar for any.
- The class of self-similar Lévy processes are called stable processes. They can be self-similar for any.
Second-order self-similarity
Definition
A wide-sense stationary process is called exactly second-order self-similar with parameter if the following hold:If instead of, the weaker condition
holds, then is called asymptotically second-order self-similar.
Connection to long-range dependence
In the case, asymptotic self-similarity is equivalent to long-range dependence.Self-similar and long-range dependent characteristics in computer networks present a fundamentally different set of problems to people doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.
Long-range dependence is closely connected to the theory of heavy-tailed distributions. A distribution is said to have a heavy tail if
One example of a heavy-tailed distribution is the Pareto distribution. Examples of processes that can be described using heavy-tailed distributions include traffic processes, such as packet inter-arrival times and burst lengths.
Examples
- The Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes.
- Ethernet traffic data is often self-similar. Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic.