Random walk visiting a cylinder infinitely often

Question

I wonder whether a $d$-dimensional random walk $S_n$, generated by the infinite i.i.d. copies of X given by:

$X=e_1=(1, 0, 0, ..., 0)$ (with probability $p_1$)

$X=e_2=(0, 1, 0, ..., 0)$ (with prob $p_2$)

. . .

$X=e_d $ (with prob $p_d$)

$X=0$ (with prob $p_0=1-p_1-p_2...-p_d$),

visits any infinite cylinder of radius $\sqrt{d}$ and parallel to the vector $\overrightarrow{p}:=(p_1, p_2,..., p_d)$ infinitely often a.s. ? More precisely, if $L$ is a line parallel to the vector $\overrightarrow{p}$ and $A:=\{ y\in \mathbb{N}^d : |y-L|\leq \sqrt{d} \}$, can I say $\mathbb{P}(S_n \in A, \; \; i.o.)=1$?

If $d=1$ this holds by law of iterated logarithm.

Answer 1

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.