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.

A continuous-time stochastic process ${\displaystyle (X_{t})_{t\geq 0}}$ is called self-similar with parameter ${\displaystyle H>0}$ if for all ${\displaystyle a>0}$, the processes ${\displaystyle (X_{at})_{t\geq 0}}$ and ${\displaystyle (a^{H}X_{t})_{t\geq 0}}$ have the same law.

A wide-sense stationary process ${\displaystyle (X_{n})_{n\geq 0}}$ is called exactly second-order self-similar with parameter ${\displaystyle H>0}$ if the following hold:

If instead of (ii), the weaker condition

holds, then ${\displaystyle X}$ is called asymptotically second-order self-similar.

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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.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