Self-similar process

Self-similar processes are stochastic processes satisfying a mathematically precise version of the self-similarity property.

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.

A continuous-time stochastic process

is called exactly second-order self-similar with parameter

is called asymptotically second-order self-similar.

, asymptotic self-similarity is equivalent to long-range dependence.

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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.

[6] Long-range dependence is closely connected to the theory of heavy-tailed distributions.

Examples of processes that can be described using heavy-tailed distributions include traffic processes, such as packet inter-arrival times and burst lengths.