The 2SRWB large-deviation rate-function ansatz

Let $\mathbf{P}^{\mathrm{2SRWB}}_n$ denote the probability measure for the 2SRWB model, and let $\mathfrak{Z}_n$ be the associated random variable. For $x\in[0,1]$, write $I^{\mathrm{2SRWB}}(x)$ for the rate function.

2SRWB rate-function ansatz. For each $x\in[0,1]$, there is a regular function $I^{\mathrm{2SRWB}}$ such that

$$\lim_{n\to\infty}n^{-1}\ln\left(\mathbf{P}^{\mathrm{2SRWB}}_n(\mathfrak{Z}_n\geq nx)\right)=-I^{\mathrm{2SRWB}}(x).$$

This is an ansatz made under the assumption that a large deviation principle holds; the paper proceeds to identify the rate function implicitly for the 2SRWB model.