Self-similar process

Distributional self-similarity

Definition

A continuous-time stochastic process (X_t){t\ge0} is called self-similar with parameter H0 if for all a0, 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\in1/2,\infty).

Second-order self-similarity

Definition

A [wide-sense stationary process (X_n){n\ge0} is called exactly second-order self-similar with parameter H0 if the following hold: :(i) \mathrm{Var}(X^{(m)})=\mathrm{Var}(X)m^{2(H-1)}, where for each k\in\mathbb N_0, X^{(m)}k = \frac 1 m \sum{i=1}^m X{(k-1)m + i}, :(ii) for all m\in\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\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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.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. A distribution is said to have a heavy tail if : \lim_{x \to \infty} e^{\lambda x}\Pr[Xx] = \infty \quad \mbox{for all } \lambda0., 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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References

1. §1.4.2 of Park, Willinger (2000)
2. Park, Willinger (2000)
3. §1.4.1 of Park, Willinger (2000)
4. (February 1994). "On the Self-similar Nature of Ethernet Traffic (Extended Version)". [[IEEE]].
5. "The Self-Similarity and Long Range Dependence in Networks Web site". Cs.bu.edu.
6. (2011-12-27). "Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality". American Physical Society (APS).
7. (2022). "Stable Lévy Processes via Lamperti-Type Representations". [[Cambridge University Press]].
8. (1994). "Stable Non-Gaussian Random Processes". Chapman & Hall.
9. (1991). "Brownian Motion and Stochastic Calculus". [[Springer Verlag]].