= Solution
Assume $\sigma^2>0$; otherwise the scaled sum is deterministic and no nontrivial <rate function> is possible. The sum has <normal distribution> $B_N\sim N(N\mu,N\sigma^2)$. For $X_N=B_N/N^{1+\varepsilon}$, use the faster <large-deviation speed> $b_N=N^{1+2\varepsilon}$. The <moment-generating function of a normal distribution> yields
$$
\frac1{b_N}\log\mathbb E e^{b_N\theta X_N}
=\mu\theta N^{-\varepsilon}+\frac{\sigma^2\theta^2}{2}
\longrightarrow \Lambda(\theta)=\frac{\sigma^2\theta^2}{2}.
$$
The limiting <cumulant-generating function> is finite and differentiable on all of $\mathbb R$, hence satisfies the <Gärtner–Ellis theorem>. Its <Legendre-Fenchel transform> is obtained by maximizing $\theta x-\sigma^2\theta^2/2$, whose maximizer is $\theta=x/\sigma^2$. Thus
$$
\boxed{\text{speed }N^{1+2\varepsilon},\qquad I_B(x)=\frac{x^2}{2\sigma^2}\quad(x\in\mathbb R).}
$$
The drift disappears because the scaled <mean> is $\mu/N^\varepsilon\to0$. The <rate function> is nontrivial and good. The corrected scaled <cumulant-generating function> used here contains an <expectation>; the corresponding formula in the paper's reference material omits it.
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