I struggle proving a proposition. Assume that
$$T=\inf \{ n > 0 : X_n \leq 0 \}$$
Where $X_{n+1}=X_n+ Y_{n+1}$ and $(Y_n)$ are independent gaussians random variables $\mathcal{N}(\mu_n, \sigma_n^2)$. I would like to find a sufficient condition such that $$\mathbb{E}(T)=\infty$$ I first tried with basic definition of expectation where $$\mathbb{P}(T=n)=\mathbb{P} \left( \bigcap_{k=0}^{n-1} \{ X_k > 0\} \cap \{ X_n \leq 0 \} \right)$$ $$\mathbb{P}(T=n)=\prod_{i=0}^{n-1} \mathbb{P}(X_0 > 0, \dots, X_k > 0 | X_{k+1} > 0, \dots, X_n \leq 0)\mathbb{P}(X_n \leq 0)$$ But I encounter difficulties since $(X_n)$ are not independent variables. If you have any advice or idea to express the expectation in a better way (I found articles on Wald's identity but can't use it on my problem), I would be extremely grateful. Thanks again for your time.