Given a sequence of i.i.d. random variables $X_1,X_2,\cdots,X_n\sim N(\mu,1)$ and define the sum (gaussian random walk) as $S_n(\mu) = \sum\limits_{i=1}^n X_i$. We want to say something about the quantity $N(\mu)$, which is defined as $$ N(\mu): =\inf\left\{n\in \mathbb N^+ : e^{-\frac{1}{n}S_n(\mu)^2}\leq \alpha\right\} = \inf\left\{n\in \mathbb N^+: |S_n|\geq \sqrt{-n\log\alpha}\right\} $$ where $\alpha = \Omega(1)\in (0,0.5)$, say $0.025$.
The distribution of $N(\mu)$ itself is untractable, but the right tail is fairly easy to be bounded via any Hoeffding-type inequality. The question is about the left tail: $$ \mathbb P(N(\mu)\leq \beta)\text{ or any thing control N is small}, $$ where $\beta$ is presumably $\beta\sim -\log \alpha \cdot \mu^{-2}$ this order.
I found this paper and reference therein study a similar case (in a fairly general framework with weaker assumptions), but they don't consider the left tail bound.
The reason why I want to consider left tail bound is to upper bound $\text{Var}(\bar{X}_N)$ or equivalently $E[1/N]$ (or something like $E[1/N^2]$).