Let $0
. A real-valued Gaussian process $\left(B_H(t)\right)_{t\geq 0}$ is called fractional Brownian motion (fBm) if $\ \mathbb{E}[B_H(t)]=0$ and $\mathbb{E}[B_H(t)B_H(s)]={\mathbb{E}[B_H(1)^2]\over2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right).$
I would like to prove that a fBm is selfsimilar with parameter $H$, and I found something in the literature:
For any $a>0$ we have \begin{eqnarray*} \mathbb{E}[B_H(at)B_H(as)] &=& {\mathbb{E}[B_H(1)^2]\over2}\left((at)^{2H}+(as)^{2H}-(a|t-s|)^{2H}\right)\\ &=& a^{2H}\mathbb{E}[B_H(t)B_H(s)]\\ &=& \mathbb{E}\left[\left(a^HB_H(t)\right)\left(a^HB_H(s)\right)\right] \end{eqnarray*} Since all processes here are mean zero Gaussian, this equality in covariance implies that $B_H(at)\overset{d}{=}a^HB_H(t)$.
I cannot really follow the conclusion of the last sentence. Any help please.