Efficiency of GLS over OLS when regressors are not fixed

Suppose we have a regression model $$y=X\beta+u,\quad E(u)=0,\quad E(uu')=\Sigma.$$ Let $\hat\beta$ and $\bar\beta$ respectively denote the OLS and GLS estimator. Then, when $X$ is fixed (or when $X$ and $u$ are independent), one can show the efficiency result $$\text{Var}(\hat\beta)=(X'X)^{-1}X'\Sigma X(X'X)^{-1}\succeq(X'\Sigma^{-1}X)^{-1}=\text{Var}(\bar\beta).$$ Is there a similar asymptotic result that holds when $X$ is stochastic? If so, could you please provide a statement and a (sketch of) proof or point me to some references?

• What does the inequality-like symbol mean? It must be related to matrices because I've always seen it in that contest, but $\mathbf {R}^{n,n}$ is not ordered. Sep 17 '16 at 16:15
• @DeltaIV $A\succeq B$ usually means $A-B$ is positive semidefinite. Sep 17 '16 at 16:37

I think you have to look at the asymptotic variances of OLS and GLS. That is, at the variances in the multivariate normal distributions in the distribution limits of $\sqrt{n}(\hat{\beta}-\beta)$ and $\sqrt{n}(\tilde{\beta}-\beta)$. Under quite general conditions, the asymptotic results state that $$\sqrt{n}(\hat{\beta}-\beta) \stackrel{d} \rightarrow \mathcal{N}\left(0, \text{Avar}(\hat{\beta})\right)$$ $$\sqrt{n}(\tilde{\beta}-\beta) \stackrel{d} \rightarrow \mathcal{N}\left(0, \text{Avar}(\tilde{\beta})\right),$$ where $$\text{Avar}(\hat{\beta}) = \left(\text{plim} \frac{1}{n}X′X\right)^{−1} \, \text{plim} \frac{1}{n}X′\Sigma X \, \left(\text{plim} \frac{1}{n}X′X\right)^{−1}$$ $$\text{Avar}(\tilde{\beta}) = \left(\text{plim} \frac{1}{n}X′\Sigma^{−1}X\right)^{−1}.$$
From here you can see that you can draw similar conclusion asymptotically: $$\text{Avar}(\hat{\beta}) \succeq \text{Avar}(\tilde{\beta}).$$