📂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.