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 (or Brownian motion) 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: (i) , where for each , (ii) for all , the autocorrelation functions and of and are equal. If instead of (ii), the weaker condition (iii) pointwise as 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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.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.§1.4.2 of Park, Willinger (2000) 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.
References Park, Willinger (2000)
§1.4.1 of Park, Willinger (2000)
Theorem 3.2 of
Chapter 2: Lemma 9.4 of
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