Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic

In a recent paper [5] it was shown that under suitable conditions stationary distributions of the (scaled) queue lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we provide an alternative proof of this result that does not require exponential integrability. The conditions that we impose (in addition to natural heavy traffic and stability assumptions) are precisely the i.i.d. and square integrability requirements on the network primitives that are typically made in heavy traffic analysis.