Self-similar process explained

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

(X _{t)} _{t\ge0} is called self-similar with parameter H> 0 if for all a> 0 , the processes (X _{at} ) _{t\ge0} and (a ^{HX} _{t)} _{t\ge0} 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[ 1 / 2 , inf ty) .

Second-order self-similarity

Definition

(X _{n)} _{n\ge0} is called exactly second-order self-similar with parameter H> 0 if the following hold:

(i)

Var (X ^{(m)} ) = Var (X)m ^{2(H-1)} , where for each k\in N _{0} , X ^{(m)} _{k} = (1)/(m) \sum ^{m} _{i =1} X _{(k-1)m} ,

(ii) for all

m\in 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 inf ty holds, then X is called asymptotically second-order self-similar.

Connection to long-range dependence

In the case

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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.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.^[1] A distribution is said to have a heavy tail if

\lim _{x} e ^{λ} \Pr[X>x] = inf ty forall λ > 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

Notes and References

1. §1.4.2 of Park, Willinger (2000)