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

File:Wiener process animated.gif

A plot of $(1/\sqrt{c})W_{ct}$ for $W$ a Brownian motion and c decreasing, demonstrating the self-similarity with parameter $H=1/2$.

Definition

A continuous-time stochastic process $(X_t{)}_{t\ge 0}$ is called self-similar with parameter $H>0$ if for all $a>0$, the processes $(X_{at}{)}_{t\ge 0}$ and $(a^H{X}_t{)}_{t\ge 0}$ have the same law.Template:R/superscript

Examples

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

Second-order self-similarity

Definition

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

(i) $\mathrm{V}\mathrm{a}\mathrm{r}(X^{(m)})=\mathrm{V}\mathrm{a}\mathrm{r}(X)m^{2(H-1)}$, where for each $k\in {\mathbb{\mathbb{N}}}_0$, $X_k^{(m)}=\frac{1}{m}\sum _{i=1}^m{X}_{(k-1)m+i},$

(ii) for all $m\in {\mathbb{\mathbb{N}}}^{+}$, the autocorrelation functions $r$ and $r^{(m)}$ of $X$ and $X^{(m)}$ are equal.

If instead of (ii), the weaker condition

(iii) $r^{(m)}\to r$ pointwise as $m\to \mathrm{\infty }$

holds, then $X$ is called asymptotically second-order self-similar.Template:R/superscript

Connection to long-range dependence

In the case $1/2<H<1$, asymptotic self-similarity is equivalent to long-range dependence.Template:R/superscript 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.Template:R/superscript

Long-range dependence is closely connected to the theory of heavy-tailed distributions. A distribution is said to have a heavy tail if

$\underset{x\to \mathrm{\infty }}{\lim }{e}^{\lambda x}\mathrm{Pr}[X>x]=\mathrm{\infty }\text{for all }\lambda >0.$

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.Template:R/superscript

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.Template:R/superscript
• Ethernet traffic data is often self-similar.Template:R/superscript Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic.Template:R/superscript

References

Sources