Well, for $d\geq 2$, the projection of $S_n$ onto a hyperplane orthogonal to $\vec{p}$ is a zero-mean $(d-1)$-dimensional random walk with bounded jumps. Therefore, the answer to your question is ''yes'' for $d\leq 3$ and ''no'' for $d\geq 4$. (That fact about zero-mean random walks is well known, but see e.g. Theorem 1.5.2 of the book "Non-homogeneous random walks" by Menshikov-Popov-Wade if you need a reference.)

Well, for $d\geq 2$, the projection of $S_n$ onto a hyperplane orthogonal to $\vec{p}$ is a zero-mean $(d-1)$-dimensional random walk with bounded jumps. Therefore, the answer to your question is ''yes'' for $d\leq 3$ and ''no'' for $d\geq 4$. (That fact about zero-mean random walks is well known, but see e.g. Theorem 1.5.2 of the book "Non-homogeneous random walks" by Menshikov-Popov-Wade if you need a reference.)

P.S. I guess that it is implicitly assumed that all $p_i$s are strictly positive --- otherwise you can just forget about some of the coordinates and effectively reduce the dimension.

Well, for $d\geq 2$, the projection of $S_n$ onto a hyperplane orthogonal to $\vec{p}$ is a zero-mean $(d-1)$-dimensional random walk with bounded jumps. Therefore, the answer to your question is ''yes'' for $d\leq 3$ and ''no'' for $d\geq 4$. (That fact about zero-mean random walks is well known, but see e.g. Theorem 4.1.1 of the book "Random walk: a modern introduction" by Lawler and Limic if you need a reference.)

Well, for $d\geq 2$, the projection of $S_n$ onto a hyperplane orthogonal to $\vec{p}$ is a zero-mean $(d-1)$-dimensional random walk with bounded jumps. Therefore, the answer to your question is ''yes'' for $d\leq 3$ and ''no'' for $d\geq 4$. (That fact about zero-mean random walks is well known, but see e.g. Theorem 1.5.2 of the book "Non-homogeneous random walks" by Menshikov-Popov-Wade if you need a reference.)