Suppose there is a random walk $S_n=\sum_{k=1}^{n}X_k,n\geq1$ and $S_0=0$, where $X_k$ are i.i.d. random variables. In addition, $-\infty< EX_1<0$ and $0<EX^2_1<+\infty$.
Define the maximum over all $S_n$ as $M:=\sup_{n\geq 0}\{S_n\}$.
It is intuitive that $M$ is almost surely $<+\infty$ and it can be verified by the Law of Large Numbers (see Bounds on random walk with a negative drift and Stopped random walks(2nd edition) by Allan Gut Theorem 10.1 in Chapter 2).
In order to derive $E(M)$ with the help of laddder variables, it is crucial to obtain the probablity that $M=0$, i.e., $S_n$ never exceeds $0$.
My question is how to obtain $P(M=0)$? It has been shown that this probability is greater than $0$ in Lemma 11.3 of Chapter 2 in Stopped random walks by Allan Gut.