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

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

Definition

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. ^[1]

Examples

• The Wiener process (or Brownian motion) is self-similar with ${\displaystyle H=1/2}$. ^[2]
• The fractional 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-similar Lévy processes are called stable processes. They can be self-similar for any ${\displaystyle H\in [1/2,\infty )}$. ^[4]

Second-order self-similarity

Definition

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:

(i) ${\displaystyle \mathrm {Var} (X^{(m)})=\mathrm {Var} (X)m^{2(H-1)}}$, 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 ${\displaystyle m\in \mathbb {N} ^{+}}$, the autocorrelation 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 ${\displaystyle m\to \infty }$

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

Connection to long-range dependence

In the case ${\displaystyle 1/2<H<1}$, asymptotic self-similarity is equivalent to long-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 of heavy-tailed distributions. ^[7] A distribution is said to have a heavy tail if

${\displaystyle \lim _{x\to \infty }e^{\lambda x}\Pr[X>x]=\infty \quad {\mbox{for all }}\lambda >0.\,}$

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. ^[8]

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. ^[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]

References

1. §1.4.1 of Park, Willinger (2000)
2. Chapter 2: Lemma 9.4 of Ioannis Karatzas; Steven E. Shreve (1991), Brownian Motion and Stochastic Calculus (second ed.), Springer Verlag, doi:10.1007/978-1-4612-0949-2, ISBN   978-0-387-97655-6
3. Gennady Samorodnitsky; Murad S. Taqqu (1994), "Chapter 7: "Self-similar processes"", Stable Non-Gaussian Random Processes, Chapman & Hall, ISBN   0-412-05171-0
4. Theorem 3.2 of Andreas E. Kyprianou; Juan Carlos Pardo (2022), Stable Lévy Processes via Lamperti-Type Representations, New York, NY: Cambridge University Press, doi:10.1017/9781108648318, ISBN   978-1-108-48029-1
5. Will E. Leland; Murad S. Taqqu; Walter Willinger; Daniel V. Wilson (February 1994), "On the Self-similar Nature of Ethernet Traffic (Extended Version)", IEEE/ACM Transactions on Networking, 2 (1), IEEE: 1–15, doi:10.1109/90.282603
6. "The Self-Similarity and Long Range Dependence in Networks Web site". Cs.bu.edu. Archived from the original on 2019-08-22. Retrieved 2012-06-25.
7. §1.4.2 of Park, Willinger (2000)
8. Park, Willinger (2000)
9. Kendal, Wayne S.; Jørgensen, Bent (2011-12-27). "Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality". Physical Review E. 84 (6). American Physical Society (APS): 066120. Bibcode:2011PhRvE..84f6120K. doi:10.1103/physreve.84.066120. ISSN   1539-3755. PMID   22304168.

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   0471319740