Variant of Cramer's Large Deviation Theorem

Let $\Lambda(t) = \log \mathbb{E}[\exp(tX_1)]$ and define $\Lambda^*(x) := \sup_{t\in\mathbb{R}} (tx - \Lambda(t))$. Then according to Cramer's theorem,

$$\lim_{n \to \infty} \frac{1}{n} \log \left( P \left( \sum_{i=1}^n X_i \geq nx \right) \right) = -\Lambda^*(x).$$

Is there a similar identity for$$\lim_{n \to \infty} \frac{1}{n} \log \left( P \left( \sum_{i=1}^n X_i \geq (n+k)x \right) \right)$$where $k$ is constant?