# Random walk visiting a cylinder infinitely often

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.

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.