= Solution
Let $\kappa=2(1-H)$ and fix $a>0$. Starting with part (b), the <contraction principle for large deviations> under the map $q\mapsto q/a$ gives an LDP for $r(X)/(aN)$ with speed $N^\kappa$ and <rate function> $q\mapsto J(aq)$.
Alternatively, replace $N$ by $aN$ in parts (a) and (b), using the real-scale justification in part (a). The same family has <large-deviation speed> $(aN)^\kappa$ and <rate function> $J(q)$. When expressed at speed $N^\kappa$, that <rate function> is $a^\kappa J(q)$. The two descriptions of the same <large deviation principle> must agree:
$$
J(aq)=a^\kappa J(q).
$$
For clarity, uniqueness here is a general property of <lower semicontinuous> <rate functions>: at a point $x$, the shrinking-neighborhood logarithmic bounds recover $I(x)$, since $\inf_{B(x,\eta)}I\to I(x)$ as $\eta\downarrow0$. Thus two <rate functions> for the same family and speed cannot differ.
Taking the base argument $q=1$ and replacing $a$ by any positive <queue> size gives
$$
\boxed{J(q)=q^{2(1-H)}J(1)\quad(q>0),\qquad J(0)=0.}
$$
For $q<0$ the <rate function> is infinite. In the usual nondegenerate case $J(1)$ is finite, so the formula also holds at $q=0$. This is the <self-similar Gaussian workload rate>; the exponent comes from the <Hurst parameter>, while the constant comes from the path <rate function> and the drift and noise scales.
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