We study the long-time dynamical properties of a chain of harmonically bound Brownian particles. This chain is allowed to wander everywhere in the plane. We show that the scaling variables for the occupation times T_j, areas A_j and winding angles θ_j (j = 1,...,n labels the particles) take the same general form as in the usual Brownian motion. We also compute the asymptotic joint laws P({T_j}), P({A_j}), P({θ_j}) and discuss the correlations occurring in those distributions.