Suppose that $X(t)$ is an $\alpha$-fractal Brownian motion.
We know that such a process has the following self-similarity property:
$$X(t) \sim \frac{1}{|a|^{\alpha}} X(at)$$
Where $\sim$ denotes having the same probability distribution.
What I want to understand is whether or not this $\sim$ preserves covariance. For example, does this mean that for any random process $Y(t)$:
$$E[X(t)Y(t)] = E[\frac{1}{|a|^{\alpha}} X(at) Y(t)]$$
The answer isn't clear to me because I'm unsure if the self-similarity means they share all properties including covariance etc, or if they just share the same distribution at time $t$ (ie. Both are $N(0,\sigma^2 t^{2\alpha})$).
You need to know that the pair $(X(t),Y(t))$ has the same distributions as the pair $(|a|^{-\alpha}X(at),Y(t))$. Your expectation identity will then follow because the expectation is determined by the joint distribution.