Theorem (Helly-Bray): \(\x_n \inD \x\) if and only if \(\Ex g(\x_n) \to \Ex g(\x)\) for all continuous bounded functions \(g: \real^d \to \real\).
Traditionally, “Helly-Bray Theorem” refers only to the forward part of the theorem.
Proof: Ferguson, A Course in Large Sample Theory (1996), Theorem 3.
See also: Portmanteau theorem, which generalizes the Helly-Bray Theorem.