Helly-bray theorem

Theorem (Helly-Bray): $x_{n} \overset{d}{⟶} x$ if and only if $E g (x_{n}) \to E g (x)$ for all continuous bounded functions $g : R^{d} \to R$ .

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.