LetX t = t i=1 i , where independent d-dimensional r.vec.s i take on 2

LetX_t =Σ^t _i=1 ξ_i, where independent d-dimensional r.vec.’s ξ_i take on 2^d vector values (±1, ...,±1) with equal probabilities 1/2^d. Another way to state the problem is to write ξ_i = (ξ_i1, ..., ξ_id), where ξ_ik’s are independent r.v.’s assuming values ±1 with equal probabilities 1/2. Let X₀ = (0, ...,0).

For d = 1, this is a classical symmetric random walk we considered repeatedly.
For d = 2, we may view it as the “concatenation” of two independent random walks: a particle moves independently in two perpendicular directions; one step “to right or left”
in each direction with equal probabilities. The set of all possible values (or states) of the process X_t may be identified with vertices of a two-dimensional lattice with integer-valued coordinates. The interpretation for the dimension d ≥ 3 is similar.
Prove that all states are recurrent for d = 1,2 and transient for d ≥ 3.