Self-similarity and long-range dependence in fractional Brownian motions
I came across this thesis and want to take a note on the concepts of self-similarity and long-range dependence which are among notable properties of fractional Brownian motion. Possibly one reason why the term “fractional” is in the name of this stochastic process is that there is a close connection between self-similarity definition and fractals.
Definition of self-similarity
In a general sense, self-similar object looks exactly or approximately similar to a part of itself (which is also mean a small scaled version the object).
This image (source: Wikipedia) is a beautiful illustration for self-similarity here.
In statistics, roughly speaking, a stochastic process is self-similar if any finite samples from the process has the same ditribution with a scaled version of aggregation (mean, first-order statistic) of the finite samples.
Mathematically, consider a discrete stochastic process $X_k$. For any $m > 1$, a new stochastic process $X^{(m)}_k$ is defined as
\[X^{(m)}_k = \frac{1}{m} (X_{km} + \dots + X_{(k+1)m - 1})\]If we take a finite sample of $X$ and scaled version of $X^{(m)}$, for example,
\[(X_{k_1}, \dots, X_{k_d}) \quad ,\quad (m^{1-H}X_{k_1}^{(m)}, \dots,m^{1-H}X_{k_d}^{(m)})\]and these two have the same distribution, we say $X$ is self-similar with Hurst parameter $H$.
Definition of long-range dependence
The long-range dependence is decided by autocovariance function $\gamma(k)$. We say a stochastic process has long-range dependence, long memory if the sum of autocovariance is unbounded,
\[\sum_{k=0}^\infty \gamma(k) = \infty\]Still, $\gamma(k)$ decays.
In most cases, to identify long-range dependence, one may consider the form
\[\gamma(k) = \mathcal{O}(\lvert k \rvert^{-\alpha})\]Note that under long-range dependence, some statistical tests may be affected as the standard deviation of the mean of $X$ is different from the estimation that does not assume long-range dependence, resulting in incorrect confidence intervals in the tests.
Fractional Brownian motion
Fractional Brownian motion, $B^H$, is an generalization of Browian motion, sharing some properties with Brownian motions such as
- Stationary increments
- Start at 0 and having zero mean
Fractional Brownian motion is also viewed as a Gaussian process with covariance
\[R^H(t, s) = \frac{1}{2}\{t^{2H} + s^{2H} - (t - s)^{2H}\}\]The noise corresponding to this is defined as $X$
\[X_k = B^H(k + 1) - B(ks)\]Fraction Brownian noise appears to exhibit both long-range dependence and self-similarity.
Self-similarity
Let’s compare between $mX^{(m)}$ and $m^HX$. These two have zero mean. The following says that they have the same covariance
\[\begin{aligned} & \text{Cov}(X_{km} + \dots, X_{(k+1)m - 1}, X_{lm} + \dots, X_{(l+1)m - 1}) \\ = & \text{Cov}(B^H((k+1)m) - B^H(km), B^H((l+1)m) - B^H(lm)) \\ = & \text{Cov}(m^H(B^H(k+1) - B^H(k)), m^H(B^H(l+1) - B^H(l))) \\ = & \text{Cov}(m^H X_k, m^H X_l) \end{aligned}\]To this point, we can say they have same distribution because they are both Gaussian, having same mean and covariance.
Long-range dependence The autocovariance in this case is
\[\gamma(k) = \frac{1}{2}(\lvert k-1 \rvert^{2H} - 2 \lvert k \rvert^{2H} + \lvert k+1 \rvert^{2H})\]we can obtain that
\[\gamma(k) = \mathcal{O}(k^{2H-2})\]It is because Taylor expansion of $h(x) = (1 - x)^{2H} - 2 + (1 + x)^{2H}$ and $\gamma(k) = \frac{1}{2}k^{2H}h(1/k)$. So $\sum \gamma(k) = \infty$ when $1/2<H< 1.$