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 H>0 if for all a>0, the processes and have the same law.

Examples

Second-order self-similarity

Definition

A wide-sense stationary process is called exactly second-order self-similar with parameter H>0 if the following hold: If instead of (ii), the weaker condition holds, then X is called asymptotically second-order self-similar.

Connection to long-range dependence

In the case 1/2<H<1, 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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.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