Assume that $S_i(k) =\sum_{t=1}^k X_i(t)$ for $i = 1,2.$ $X_i(t)$ are i.i.d. random variables with positive mean. What is the probability that $\inf_k \{\max_{i} S_i(k)\}< -a$, for some a > 0?
I suppose as $a\to+\infty$, $P(\inf_k \{\max_i S_i(k\} < -a) = P(\inf_k S_1(k)<-a)\cdot P(\inf_k S_2(k)<-a)$. Is that correct? How to prove it?
Partial Answer showing that your product claim is not correct, but not providing the correct answer $$ \textstyle \text{Saying}\quad \inf_k \max_i S_i(k) \le -a \quad\text{means}\quad \text{there exists a $k$ so that $S_1(k) \le -a$ and $S_2(k) \le -a$}. $$ So we do not simply take the product: we need $S_1$ and $S_2$ to be below $-a$ at the same time. Taking the product would just say "both $S_1$ and $S_2$ individually go below $-a$ at some point".
Note also that you have said "as $a \to \infty$", and then written $a$ on both sides. Do you mean that the two limits are the same? (For a sequence $x_n$, the limit $\lim_n x_n$ must be independent of $n$, if it exists, since $n$ is only a dummy variable.)