By Section 5.5.2 of Serfling (1980) and the fact that Γ$_{0}$ is P$_{0}$ degenerate and uniformly bounded, n$_{U}$$_{n}$ Γ$_{0}$ ⇝ ∑ ∞ k $_{=1}$λ$_{k}$ ( Z 2 k − 1). We now prove that all of the eigenvalues of h ( o ) ↦→ E$_{P}$$_{0}$ $^{[}$˜ Γ$_{0}$( O, o ) h ( O ) ] are nonnegative. Consider a submodel { P$_{t}$ : t } with first-order score h$_{1}$ ∈ L $^{2}$( P$_{0}$ ) and second-order score h$_{2}$ ≡ 0. By the second-order pathwise differentiability of Ψ, The left-hand side is nonnegative for all t = 0 since ψ$_{t}$ ≥ 0 = ψ$_{0}$ under H$_{0}$ . Thus taking the limit inferior as t → 0 of both sides shows that Using that ˜ Γ$_{0}$ = Γ$_{0}$ under H$_{0}$ and Γ$_{0}$ = 1 $_{2}$D Ψ 2 , we have that 〈 o ↦→ E$_{P}$$_{0}$ [ ˜ Γ$_{0}$( O, o ) h$_{1}$ ( O )] , h$_{1}$ 〉 ≥ 0, where the inner product is that of L $^{2}$( P$_{0}$ ). For any h$_{1}$ ∈ L $^{2}$( P$_{0}$ ), it is well known that one can choose a submodel P$_{t}$ with first-order score h$_{1}$ ∈ L $^{2}$( P$_{0}$ ). Hence the above relation holds for all h$_{1}$ ∈ L $^{2}$( P$_{0}$ ) and all of the eigenvalues of h ( o ) ↦→ E$_{P}$$_{0}$ $^{[}$˜ Γ$_{0}$( O, o ) h ( O ) ] are nonnegative. Proof of 2. In this case Γ$_{0}$( o$_{1}$, o$_{2}$ ) = 2 D R 0 ( o$_{1}$ ) D R 0 ( o$_{2}$ ) under H$_{0}$ . The central limit theorem yields that σ − 1 1 √ n ( P$_{n}$ − P$_{0}$ ) D R 0 ⇝ Z . By the continuous mapping theorem, σ − 2 1 n ( P$_{n}$ − P$_{0}$ ) $^{2}$Γ$_{0}$ / 2 ⇝ Z $^{2}$. Now use that The above quantity converges in distribution to Z 2 − 1 by the weak law of large numbers and Slutsky’s theorem. 54 n$_{U}$$_{n}$ Γ$_{0}$ 2 σ 2 1 = n 2 σ 2 $_{1}$( n − 1) [ n ( P$_{n}$ − P$_{0}$ ) $^{2}$Γ$_{0}$ − 1 n n ∑ i =1 Γ$_{0}$( O$_{i}$, O$_{i}$ )$^{]}$ = n 2 σ 2 $_{1}$( n − 1) [ n ( P$_{n}$ − P$_{0}$ ) $^{2}$Γ$_{0}$ − 2 n n ∑ i =1 D R 0 ( O$_{i}$ ) 2 ] . 1 2 ∫ ∫ D Ψ 2 ( o$_{1}$, o$_{2}$ ) h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) dP$_{0}$ ( o$_{2}$ ) ≥ 0 . ψ$_{t}$ − ψ$_{0}$ t 2 = 1 2 ∫ ∫ D Ψ 2 ( o$_{1}$, o$_{2}$ ) h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) dP$_{0}$ ( o$_{2}$ ) + o (1) .