Let ($S_n^{(1)}$)n≥0, . . . ,($S_n^{(1)}$ (1))n≥0 be independent symmetric random walks on the integers, each starting at 0. Consider the RW Sn = ($S_n^{(1)}$ , . . . , $S_n^{(1)}$ ) on the lattice $Z^d$. Show the dimensions d in which this walk is recurrent and in which transient.
So i can prove that the simple random walk on the two dimensional grid is recurrent but on the three dimensional grid it is transient, meaning a drunk person cant return to his home on 3D but on 2D with Pr=1 with infinite wait time. Now we have Sn set of RWs, is it simply the same as its elements sinec theyre independent? i am a lil confused over what Sn is btw, Sn is a random walk of RWs, or it sipmly means d dimension RWs? Any hints would be appreciated.