Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_1)$ where $t\in [0,\pi]$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $s$ such that $X_t$ is strictly positive with high probability? i.e.

$$P(\inf X_t>0)\rightarrow 1.$$

What I tried: I've already found covering consitituted by uniformly sampled points in the region $T$ as the centers of the balls with $\epsilon$ radius. Also, on the random centers, I prove that $P(X_t>0)\rightarrow 1$ where $t\in N$ and $s<\sqrt{n}$.

Then I would like to use epsilon net argument to deal with the points that are not on the net \begin{equation} \begin{aligned} P(\inf_{t\in T} X_t>0)&=1-P(\inf_{t\in T} X_t<0)\\&=1-P(\inf_{t\in T} (X_t-X_{\pi(t)}+X_{\pi(t)})<0)\\&\geq 1-P(\inf_{t\in T} (X_t-X_{\pi(t)})<0)+P(\inf X_{\pi(t)}<0)\\ \end{aligned} \end{equation}

The second probability term $P(\inf X_{\pi(t)}<0)$ can be lower bounded by union bound.

However, I was stuck on bounding the first probability term.

Normally what we did is $P(\sup_{t\in T} X_t-X_{\pi(t)}>0)\leq P(\sup_{t\in T}|X_t-X_{\pi(t)}|>0)$ and assuming the process is Lipschitz, i.e. $|X_t-X_{\pi(t)}|\leq C|t-\pi(t)|$ where $C$ is a random variable, then the probability can be further bounded by $P(C\epsilon>0)$.

Following the same procedure here, assuming we have $|X_t-X_{\pi(t)}|<C|t-\pi(t)|$. Then we have the lower bound of the first term $$P(\inf_{t\in T} (X_t-X_{\pi(t)})<0)\leq P(\inf_{t\in T} C|t-\pi(t)|>0).$$

Clearly, this bound is $1$. Then we would have no chance to prove the first probability goes to $0$ when $n\rightarrow \infty$. I think the issue is that we lost too much when doing $X_t-X_{\pi(t)}<C|t-\pi(t)|$.

Following this book, I also tried to look into chaining (dealing with probabilistic Lipschitz bound) and slicing (dealing with $E(X_t)\neq 0$ case). But I think the issue can not be resolved by applying these more technical methods.