Covering problems

For a simple symmetric random walk on the lattice $\mathbf{Z}^d$, let $S_n=X_1+\cdots+X_n$ and let $X_1,X_2,\ldots$ be a sequence of independent and identically distributed random vectors with
$$ \mathbf{P}\{X_1=e_i\}=\mathbf{P}\{X_i=-e_i\}=\frac{1}{2d}\qquad (i=1,2,\ldots,d), $$
where $e_1,e_2,\ldots,e_d $ are the orthogonal unit vectors of $\mathbf{Z}^d$. Denote by $R_d (n)$ the radius of the largest ball $\{x\in\mathbf{Z}^d:\|x\|\le r\}$ every point of which is visited at least once in time $n$.The present paper studies the limiting behavior of $R_d (n)$ for $d=1$, $d=2$, and $d\ge3$.