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I am asked in a homework question to prove asymptotic normality for the generalized method of moments estimator. The assumptions (which i think are necessary to solve this particular subproblem) given in the theorem are

  1. $ (Z_i)_{i \in \mathbb{N}}$ is a sequence of i.i.d. random variables.
  2. $g(z|\theta)$ is continuously differentiable wrt. $\theta$ in a neighborhood $\mathcal{N}$ of $\theta_0\in Int(\Theta)$ ($g$ is a moment restriction function, $\theta_0$ is the true parameter, and $\Theta\subset\mathbb{R}^k$ is the parameter space)
  3. $\mathbb{E}[\sup_{\theta\in \mathcal{N}}||g(Z_i|\theta)||]<\infty $

In the concluding argument of the proof I need to show that $G_n(\theta):=\frac{1}{n}\sum_{i=1}^{n}\partial_\theta g(Z_i|\theta)$ converges uniformly to $G(\theta) = \mathbb{E}[\partial _\theta g(Z_i|\theta)]$, i.e. $$\sup_{\theta\in\mathcal{N}} ||G_n(\theta) - G(\theta)||\stackrel{P}{\rightarrow}0$$ It is hinted at that the convergence follows from conditions 2 and 3. I have also snooped around various stack exchanges and gotten a hunch that the Borel-Cantelli lemma might be helpful. But at this point I am truly lost.

Any help would be greatly appreciated!

(If you feel like you would need more information on the problem, please let me know)

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1 Answer 1

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Conditions 2. and 3. are essentially the conditions for the Uniform Law of Large Numbers. In particular, condition 3. is the compact way to express the dominance condition of the ULLN.

A literature pointer to start is Newey, W. K., & McFadden, D. (1994). Large sample estimation and hypothesis testing. Handbook of econometrics, 4, 2111-2245. Lemma 2.4,

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