Let {*v*_{n}(*θ*)} be a sequence of statistics such that when*θ* =*θ*_{0},*v*_{n}(*θ*_{0})
$$\mathop \to \limits^D $$
*N*_{p}(0,*Σ*), where*Σ* is of rank*p* and*θ* ε*R*^{d}. Suppose that under*θ* =*θ*_{0}, {*Σ*_{n}} is a sequence of consistent estimators of*Σ*. Wald (1943) shows that*v*_{n}^{T}
(*θ*_{0})*Σ*_{n}^{−1}*v*_{n}(*θ*_{0})
$$\mathop \to \limits^D $$
*x*^{2}(*p*). It often happens that*v*_{n}(*θ*_{0})
$$\mathop \to \limits^D $$
*N*_{p}(0,*Σ*) holds but*Σ* is singular. Moore (1977) states that under certain assumptions*v*_{n}^{T}
(*θ*_{0})*Σ*_{n}^{−}*v*_{n}(*θ*_{0})
$$\mathop \to \limits^D $$
*x*^{2}(*k*), where*k* = rank (*Σ*) and*Σ*_{n}^{−}
is a generalized inverse of*Σ*_{n}. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (*Σ*_{n}) =*k* for*n* sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions.