SelfSimiliarGaussianProcess - crowlogic/arb4j GitHub Wiki

Self-similarity is a property of certain Gaussian processes where the process looks statistically similar to itself when scaled in time. A classic example of a self-similar Gaussian process is fractional Brownian motion (fBm).

To prove the self-similarity of a Gaussian process, you typically need to show that for any positive constant $a$, the process $X(t)$ has the same finite-dimensional distributions as $a^H X(a^{-1} t)$, where $H$ is the Hurst parameter.

Here's a general outline of how you might prove this for a Gaussian process:

1. Definition: Start by defining the Gaussian process you're interested in. For instance, fractional Brownian motion $B_H(t)$ is a Gaussian process with mean 0 and covariance function given by:

R(t, s) = \frac{1}{2} (|t|^{2H} + |s|^{2H} - |t-s|^{2H})

2. Scaling Property: Consider the process scaled by a factor $a$:

Y(t) = a^H X(a^{-1} t)

Compute the mean and covariance function of $Y(t)$.

3. Comparison: Compare the mean and covariance function of $Y(t)$ with that of $X(t)$. If they are the same, then the process is self-similar with parameter $H$.

For the case of fractional Brownian motion:

1. The covariance of the scaled process $Y(t)$ is:

R_Y(t, s) = E[Y(t)Y(s)] = E[a^H B_H(a^{-1} t) a^H B_H(a^{-1} s)] = a^{2H} R(a^{-1} t, a^{-1} s)

2. Plugging in the covariance function for $B_H(t)$, we get:

R_Y(t, s) = a^{2H} \frac{1}{2} (|a^{-1} t|^{2H} + |a^{-1} s|^{2H} - |a^{-1} t - a^{-1} s|^{2H})

= \frac{1}{2} (|t|^{2H} + |s|^{2H} - |t-s|^{2H})

Which is the same as the covariance function for $B_H(t)$.

3. Since the mean is 0 for both processes and the covariance functions are the same, $B_H(t)$ is self-similar with parameter $H$.

This is a very high-level overview, and the details can get quite technical depending on the Gaussian process in question. If you're looking to prove self-similarity for a specific Gaussian process, you'll need to delve into the specifics of that process.