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Self-similarity and the Hurst Index of Stochastic Processes 📂Probability Theory

Self-similarity and the Hurst Index of Stochastic Processes

Definition 1 2

A stochastic process $\left\{ X_{t} \right\}$ is said to be $H$-self-similar if for all $a > 0$, it satisfies the following equation. $$ X_{at} \overset{D}{=} a^{H} X_{t} $$ Here, $\overset{D}{=}$ denotes equality in distribution, and the parameter $H>0$ is referred to as the Hurst Index.

Example

Considering the Brownian motion $W_{t}$, where $W_{t} \sim N(0,t)$ applies. For instance, regarding a random variable $Z$ that follows a normal distribution $N(0,1)$, as per $a Z \sim N \left( 0, a^{2} 1 \right)$, multiplying the variance by a positive number yields taking the square root once it comes out. Therefore, $$ W_{at} \overset{D}{=} \sqrt{a} W_{t} = a^{1/2} W_{t} $$ it can be said that Brownian motion possesses $H$-self-similarity.


  1. Yang. (2008). LRD of Fractional Brownian Motion and Application in Data Network: p5. ↩︎

  2. Sottinen. (2003). Fractional Brownian Motion in Finance and Queueing: p6. ↩︎