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 A continuous-time stochastic process ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} is called self-similar with parameter H > 0 {\displaystyle H>0} if for all a > 0 {\displaystyle a>0} , the processes ( X a t ) t ≥ 0 {\displaystyle (X_{at})_{t\geq 0}} and ( a H X t ) t ≥ 0 {\displaystyle (a^{H}X_{t})_{t\geq 0}} have the same law.
Examples The Wiener process (or Brownian motion) is self-similar with H = 1 / 2 {\displaystyle 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 ∈ ( 0 , 1 ) {\displaystyle H\in (0,1)} . The class of self-similar Lévy processes are called stable processes. They can be self-similar for any H ∈ [ 1 / 2 , ∞ ) {\displaystyle H\in [1/2,\infty )} .
Second-order self-similarity
Definition A wide-sense stationary process ( X n ) n ≥ 0 {\displaystyle (X_{n})_{n\geq 0}} is called exactly second-order self-similar with parameter H > 0 {\displaystyle H>0} if the following hold:
(i) V a r ( X ( m ) ) = V a r ( X ) m 2 ( H − 1 ) {\displaystyle \mathrm {Var} (X^{(m)})=\mathrm {Var} (X)m^{2(H-1)}} , where for each k ∈ N 0 {\displaystyle k\in \mathbb {N} _{0}} , X k ( m ) = 1 m ∑ i = 1 m X ( k − 1 ) m + i , {\displaystyle X_{k}^{(m)}={\frac {1}{m}}\sum _{i=1}^{m}X_{(k-1)m+i},}
(ii) for all m ∈ N + {\displaystyle m\in \mathbb {N} ^{+}} , the autocorrelation functions r {\displaystyle r} and r ( m ) {\displaystyle r^{(m)}} of X {\displaystyle X} and X ( m ) {\displaystyle X^{(m)}} are equal. If instead of (ii), the weaker condition
(iii) r ( m ) → r {\displaystyle r^{(m)}\to r} pointwise as m → ∞ {\displaystyle m\to \infty }
holds, then X {\displaystyle X} is called asymptotically second-order self-similar.
Connection to long-range dependence In the case 1 / 2 < H < 1 {\displaystyle 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. A distribution is said to have a heavy tail if
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