= Self-similar Gaussian workload rate
{title2=$J(q)=q^{2(1-H)}J(1)$}
Let a centered <Gaussian process> $Z$ satisfy $Z(a\,\cdot)\overset d=a^H Z$ with $0<H<1$, and suppose $Z/\sqrt L$ has a <large deviation principle> in the <sublinear path space for fluid queues>, with speed $L$ and <good rate function> $I$. For $\delta,\sigma>0$, put $r=\sup_{t\geq0}(\sigma Z(t)-\delta t)$ and $\kappa=2(1-H)$. Self-similarity gives $Z(N\,\cdot)/N\overset d=Z/\sqrt{N^\kappa}$. Time substitution and the <contraction principle for large deviations> therefore give an LDP for $r/N$ with speed $N^\kappa$ and
$$
J(q)=\inf\{I(f):\sup_{t\geq0}(\sigma f(t)-\delta t)=q\}.
$$
The same family $r/(aN)$ has <rate function> $J(aq)$ by contraction, and $a^\kappa J(q)$ by replacing $N$ with $aN$ and expressing the speed again as $N^\kappa$. Uniqueness of a <rate function> gives $J(aq)=a^\kappa J(q)$, hence $J(q)=q^\kappa J(1)$ for $q>0$, with $J(0)=0$.
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