 Self-similar Gaussian workload rate

Let a centered Gaussian process $Z$ satisfy $Z(a\cdot )\stackrel{d}{=}{a}^{H}Z$ 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 $r={\sup }_{t\ge 0}(\sigma Z(t)-\delta t)$ and $\kappa =2(1-H)$. Self-similarity gives $Z(N\cdot )/N\stackrel{d}{=}Z/\sqrt{N^{\kappa }}$. Time substitution and the contraction principle for large deviations therefore give an LDP for $r/N$ with speed $N^{\kappa }$ andThe 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 $J(0)=0$.