self-similar process

File:Wiener process animated.gif

A continuous-time stochastic process $(X\_t)\_\{t\backslash ge0\}$ is called self-similar with parameter $H>0$ if for all $a>0$, the processes $(X\_\{at\})\_\{t\backslash ge0\}$ and $(a^HX\_t)\_\{t\backslash ge0\}$ have the same law.{{r|pw141}}

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

A wide-sense stationary process $(X\_n)\_\{n\backslash ge0\}$ is called exactly second-order self-similar with parameter $H>0$ if the following hold:

:(i) $\backslash mathrm\{Var\}(X^\{(m)\})=\backslash mathrm\{Var\}(X)m^\{2(H-1)\}$, where for each $k\backslash in\backslash mathbb\; N\_0$, $X^\{(m)\}\_k\; =\; \backslash frac\; 1\; m\; \backslash sum\_\{i=1\}^m\; X\_\{(k-1)m\; +\; i\},$

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

If instead of (ii), the weaker condition

:(iii) $r^\{(m)\}\; \backslash to\; r$ pointwise as $m\backslash to\backslash infty$

holds, then $X$ is called asymptotically second-order self-similar.{{r|ltww94}}

In the case

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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.{{r|barford}}

Long-range dependence is closely connected to the theory of heavy-tailed distributions.§1.4.2 of Park, Willinger (2000) A distribution is said to have a heavy tail if

:$$

\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.{{r|ParkWillinger}}

• 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.{{r|Kendal2011b}}
• Ethernet traffic data is often self-similar.{{r|ltww94}} Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic.{{r|ParkWillinger}}

{{reflist | refs =

Park, Willinger (2000)

§1.4.1 of Park, Willinger (2000)

{{citation|author1=Will E. Leland|author2=Murad S. Taqqu|author3=Walter Willinger|author4=Daniel V. Wilson|title=On the Self-similar Nature of Ethernet Traffic (Extended Version)|journal=IEEE/ACM Transactions on Networking|volume=2|number=1|date=February 1994|doi=10.1109/90.282603|publisher=IEEE|pages=1{{ndash}}15}}

{{cite web|url=http://www.cs.bu.edu/pub/barford/ss_lrd.html |title=The Self-Similarity and Long Range Dependence in Networks Web site |publisher=Cs.bu.edu |accessdate=2012-06-25| archive-url=https://web.archive.org/web/20190822195500/https://www.cs.bu.edu/pub/barford/ss_lrd.html|archive-date=2019-08-22}}

{{cite journal | last1=Kendal | first1=Wayne S. | last2=Jørgensen | first2=Bent | title=Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality | journal=Physical Review E | publisher=American Physical Society (APS) | volume=84 | issue=6 | date=2011-12-27 | issn=1539-3755 | doi=10.1103/physreve.84.066120 | page=066120| pmid=22304168 | bibcode=2011PhRvE..84f6120K | url=https://portal.findresearcher.sdu.dk/da/publications/7f80e772-1f87-4ff3-8b07-8e38494cc650 }}

Theorem 3.2 of {{citation|author1=Andreas E. Kyprianou|author2=Juan Carlos Pardo|title=Stable Lévy Processes via Lamperti-Type Representations|publisher=Cambridge University Press|location=New York, NY|year=2022|isbn=978-1-108-48029-1|doi=10.1017/9781108648318}}

{{citation|author1=Gennady Samorodnitsky|author2=Murad S. Taqqu|title=Stable Non-Gaussian Random Processes|chapter=Chapter 7: "Self-similar processes"|publisher=Chapman & Hall|year=1994|isbn=0-412-05171-0}}

Chapter 2: Lemma 9.4 of {{citation|author1=Ioannis Karatzas|author2=Steven E. Shreve|title=Brownian Motion and Stochastic Calculus|publisher=Springer Verlag|year=1991|edition=second|isbn=978-0-387-97655-6|doi=10.1007/978-1-4612-0949-2}}

}}

Sources

• {{citation

| author1 = Kihong Park

| author2 = Walter Willinger

| isbn = 0471319740

| location = New York, NY, USA

| publisher = John Wiley & Sons, Inc.

| title = Self-Similar Network Traffic and Performance Evaluation

| year = 2000

| doi = 10.1002/047120644X}}

{{Stochastic processes}}

Category:Stochastic processes

Category:Teletraffic

Category:Autocorrelation

Category:Scaling symmetries