Images and Level Sets of Additive Random Walks

X(n) = X1(n1) + · · ·+XN (nN ), n ∈ N0 . Here, nj denotes the jth coordinate of n ∈ N0 . We are following the standard convention of starting (ordinary) random walks at the origin. That is, Xj(0) = 0 for all j = 1, . . . , N . From now on, N will always denote the temporal dimension and d the spatial one, to use the standard language of stochastic processes. In order to describe a natural partial order on Z , we define the partial order ‘4’ on Z by: for all m, n ∈ Z , m 4 n if and only if mj 6nj for all j = 1, . . . , N . This partial order induces a minimum (maximum, resp.) operator f (g, resp.): for all m, n ∈ Z , m f n (m g n, resp.) denotes the point in Z whose jth coordinate is min{mj , nj} (max{nj , mj}, resp.) for all j = 1, . . . , N .