Theorem: Suppose $X_{1}, \dots, X_{n}$ are iid with density $p (x | θ^{*})$ and that derivatives up to second order can be passed under the integral sign in $\int d P (x | θ)$ . Then

\frac{1}{\sqrt{n}} u (θ^{*}) \overset{d}{⟶} N (0, I (θ^{*}))

Proof:

\begin{aligned} V u (θ^{*}) = I (θ^{*}) & Def I \\ \sqrt{n} {\bar{u} (θ^{*}) - E u (θ^{*})} \overset{d}{⟶} N (0, I (θ^{*})) & CLT (iid) \\ \frac{1}{\sqrt{n}} u (θ^{*}) \overset{d}{⟶} N (0, I (θ^{*})) & E u (θ^{*}) = 0; \bar{u} = u / n \end{aligned}

Corollaries

The above result can be stated in a variety of ways.

Corollary (#1): Under the same conditions as the original theorem, if $I (θ^{*})$ is positive definite, then

\frac{1}{\sqrt{n}} I^{- 1 / 2} (θ^{*}) u (θ^{*}) \overset{d}{⟶} N (0, I) .

Proof: Continuous mapping theorem.

Corollary (#2): Under the same conditions as the first corollary,

I_{n} (θ^{*})^{- 1 / 2} u (θ^{*}) \overset{d}{⟶} N (0, I) .

Proof:

\begin{aligned} 1 & \frac{1}{n} I_{n} (θ^{*}) \overset{p}{⟶} & I (θ^{*}) & LLN (iid) \\ I_{n} (θ^{*})^{- 1 / 2} u (θ^{*}) & = [n \frac{1}{n} I_{n} (θ^{*})]^{- 1 / 2} u (θ^{*}) \\ = \underset{A}{\underset{⏟}{[\frac{1}{n} I_{n} (θ^{*})]^{- 1 / 2} I (θ^{*})^{1 / 2}}} \underset{b}{\underset{⏟}{\frac{1}{\sqrt{n}} I (θ^{*})^{- 1 / 2} u (θ^{*})}} \end{aligned}

At this point, note that $A \overset{p}{⟶} I$ by $1$ and CMT, while $b \overset{d}{⟶} N (0, I)$ by our first corollary. Thus, the result

I_{n} (θ^{*})^{- 1 / 2} u (θ^{*}) \overset{d}{⟶} N (0, I) .

holds by Slutsky’s theorem.

Finally, note that Corollary #2 holds for any consistent estimator of $I (θ^{*})$ . For example, if $\frac{1}{n} I_{n} (\hat{θ}) \overset{p}{⟶} I (θ^{*})$ , we would also have

I_{n} (\hat{θ})^{- 1 / 2} u (θ^{*}) \overset{d}{⟶} N (0, I)

by the same proof.