A large-deviation principle for non-parametric estimators of Loynes' exponent

Let $X(1),\ldots,X(n)$ be a sequence of consecutive increments, or alternatively let $W(1),\ldots,W(n)$ be observed workloads, and let $\theta^*$ denote Loynes' exponent. Consider estimates $\{\theta^*(n)\}$ constructed without further knowledge of the process $\{X(n)\}$. Large-deviation estimation conjecture. In broad generality, under conditions similar to those in Theorem, one can construct estimates $\{\theta^*(n)\}$ satisfying a large deviation principle: for every Borel set $B$, there is a good rate function $J:[0,\infty]\mapsto[0,\infty]$ such that

$$-\inf_{x\in B^\circ}J(x)\leq \liminf_{n\to\infty}\frac{1}{n}\log P(\theta^*(n)\in B)\leq \limsup_{n\to\infty}\frac{1}{n}\log P(\theta^*(n)\in B)\leq -\inf_{x\in \overline{B}}J(x).$$

Here $B^\circ$ and $\overline{B}$ are respectively the interior and closure of $B$, and $J$ is lower semicontinuous with compact level sets. Moreover, the estimates can be chosen consistently, meaning that $J(x)=0$ if and only if $x=\theta^*$. This conjecture concerns the statistical limits of estimating the exponential tail exponent of Loynes' distribution from observed increments or workloads; the source does not establish the conjecture or provide evidence of its resolution.