Let $\{X_i\}$ be a collection of i.i.d. Exp$(1)$ random variables. For $k \in \mathbb{Z}_{>0}$, define $$S_k = \sum_{i=1}^k X_i$$ and note that $\mathbb{E}[S_k] = k$.

I was wondering if there is any way or reference for studying the asymptotic of $$\mathbb{P}\Big(\min_{1\leq k \leq n} \tfrac{S_k - k}{\sqrt{k}} \geq \sqrt n\Big)?$$

In particular, I suspect that this probably decays like $e^{-Cn}$. When $n$ is large, this probability for fixed small $k\ll n$ decays like $e^{-C\sqrt n}$ because the steps are exponential. For large $n$ and $k$, because of CLT, we have $\mathbb{P}(\mathcal{N}(0,1) \geq \sqrt n) \leq e^{-Cn}$. Thus, $e^{-Cn}$ will dominate the decay.

I am interested in obtaining the leading order constant in the exponential, i.e. $$\lim_{n \to \infty} \frac{1}{n} \log \, \mathbb{P}\Big(\min_{1\leq k \leq n} \tfrac{S_k - k}{\sqrt{k}} \geq \sqrt n\Big).$$

Let $\{X_i\}$ be a collection of i.i.d. Exp$(1)$ random variables. For $k \in \mathbb{Z}_{>0}$, define $$S_k = \sum_{i=1}^k X_i$$ and note that $\mathbb{E}[S_k] = k$.

I was wondering if there is any way or reference for studying the asymptotic of $$\mathbb{P}\Big(\min_{1\leq k \leq n} \tfrac{S_k - k}{\sqrt{k}} \geq \sqrt n\Big)?$$

In particular, I suspect that this probably decays like $e^{-Cn}$. When $n$ is large, this probability for fixed small $k\ll n$ decays like $e^{-C\sqrt n}$ because the steps are exponential. For large $n$ and $k$, because of CLT, we have $\mathbb{P}(\mathcal{N}(0,1) \geq \sqrt n) \sim e^{-Cn}$. Thus, $e^{-Cn}$ will dominate the decay.

I am interested in obtaining the leading order constant in the exponential, i.e. $$\lim_{n \to \infty} \frac{1}{n} \log \, \mathbb{P}\Big(\min_{1\leq k \leq n} \tfrac{S_k - k}{\sqrt{k}} \geq \sqrt n\Big).$$