= Self-similarity of a stochastic process
{title2=$Z(a\,\cdot)\overset d=a^HZ$}
= Self-similar stochastic process
{synonym}
A process $Z$ is self-similar with exponent $H$ if $Z(a\,\cdot)$ and $a^HZ$ 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: $Z(N\,\cdot)/N\overset d=N^{H-1}Z$. Consequently a small-noise <large deviation principle> with inverse-variance speed produces the <large-deviation speed> $N^{2(1-H)}$ for those rescaled paths.
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