Consider the first $2n$ steps of a simple random walk on the integers, starting at the origin. A simple binomial argument shows that regardless of $n$, the origin gets visited the most (in expectation).

**Question 1:** Does this fact generalize? that is, suppose we have a symmetric random walk on $\mathbb{Z}$ starting at $0$. Is it still true that the origin gets visited the most (in expectation)?

**Question 2:** Suppose we have two random walks (not necessarily symmetric) $X=(X_j)_j$ and $Y=(Y_j)_j$ on $\mathbb{Z}_{\geq 0}$ that can be coupled such that $X_j\leq Y_j$ for all $j$, and such that the integer most visited by $Y$ is $0$. Does this imply the same for $X$? that is, is it necessarily true that the integer most visited by $X$ is $0$?

Thanks!