Self-similarity of a stochastic process ($Z(a\cdot )\stackrel{d}{=}{a}^{H}Z$)

A process $Z$ is self-similar with exponent $H$ if $Z(a\cdot )$ and $a^{H}Z$ have the same finite-dimensional distributions for every $a>0$, or the same path law when the process is considered as a random path. Fractional Brownian motion has this property with its Hurst exponent. It converts dilation of time to deterministic scaling of amplitude: $Z(N\cdot )/N\stackrel{d}{=}{N}^{H-1}Z$. Consequently a small-noise large deviation principle with inverse-variance speed produces the large-deviation speed $N^{2(1-H)}$ for those rescaled paths.