Self-similar processes arestochastic processes satisfying a mathematically precise version of theself-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 arerandom 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.

Acontinuous-time stochastic process${\displaystyle (X_t{)}_{t\ge 0}}$ is calledself-similar with parameter${\displaystyle H>0}$ if for all${\displaystyle a>0}$, the processes${\displaystyle (X_{at}{)}_{t\ge 0}}$ and${\displaystyle (a^H{X}_t{)}_{t\ge 0}}$ have the samelaw.^[1]

• TheWiener process (orBrownian motion) is self-similar with${\displaystyle H=1/2}$.^[2]
• Thefractional Brownian motion is a generalisation of Brownian motion that preserves self-similarity; it can be self-similar for any${\displaystyle H\in (0,1)}$.^[3]
• The class of self-similarLévy processes are calledstable processes. They can be self-similar for any${\displaystyle H\in [1/2,\mathrm{\infty })}$.^[4]

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

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

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

If instead of (ii), the weaker condition

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

holds, then${\displaystyle X}$ is calledasymptotically second-order self-similar.^[5]

In the case, asymptotic self-similarity is equivalent tolong-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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.^[6]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 ofheavy-tailed distributions.^[7] A distribution is said to have a heavy tail if

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

One example of a heavy-tailed distribution is thePareto distribution. Examples of processes that can be described using heavy-tailed distributions include traffic processes, such as packet inter-arrival times and burst lengths.^[8]

• TheTweedie convergence theorem can be used to explain the origin of thevariance to mean power law,1/f noise andmultifractality, features associated with self-similar processes.^[9]
• Ethernet traffic data is often self-similar.^[5] Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic.^[8]

Sources

• Kihong Park; Walter Willinger (2000),Self-Similar Network Traffic and Performance Evaluation, New York, NY, USA: John Wiley & Sons, Inc.,doi:10.1002/047120644X,ISBN 0-471-31974-0

• Stochastic processes
• Teletraffic
• Autocorrelation
• Scaling symmetries