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 $(X_t)_$ is called ''self-similar'' with parameter $H>0$ if for all $a>0$, the processes $(X_)_$ and $(a^HX_t)_$ have the same law.

Examples

*The Wiener process (or Brownian motion) is self-similar with $H=1/2$.
*The fractional Brownian motion is a generalisation of Brownian motion that preserves self-similarity; it can be self-similar for any $H\in(0,1)$.
*The class of self-similar Lévy processes are called stable processes. They can be self-similar for any H\in /2,\infty).

Second-order self-similarity

Definition

A wide-sense stationary process $(X_n)_$ is called ''exactly second-order self-similar'' with parameter $H>0$ if the following hold:
:(i) $\mathrm(X^)=\mathrm(X)m^$, where for each $k\in\mathbb N_0$, $X^_k = \frac 1 m \sum_^m X_,$
:(ii) for all $m\in\mathbb N^+$, the Autocorrelation#Definition_for_wide-sense_stationary_stochastic_process">autocorrelation functions $r$ and $r^$ of $X$ and $X^$ are equal.
If instead of (ii), the weaker condition
:(iii) $r^ \to r$ pointwise as $m\to\infty$
holds, then $X$ is called ''asymptotically second-order self-similar''.

Connection to long-range dependence

In the case 1/2, 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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.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
:
\lim_ e^\Pr >x= \infty \quad \mbox \lambda>0.\,

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

Sources
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{{Stochastic processes

Stochastic processes
Teletraffic
Autocorrelation
Scaling symmetries