# Asymptotic Normality for GEE Parameters

In the famous Liang and Zeger 1986 paper on GEEs https://www.jstor.org/stable/2336267?seq=9, they sketch a proof using the standard m-estimator arguments: (unstated) regularity conditions + first-order Taylor expansion + somehow a vanishing remainder in probability (covered by regularity conditions) + routine convergence arguments to prove asymptotic normality at rate $$\sqrt{n_c}$$, where $$n_c$$ is the number of clusters. What version of the Central Limit Theorem (CLT) for independent data is typically cited to enable asymptotic normality here? I feel like I never see this pointed out specifically anywhere.

To elaborate a bit further, let $$\varphi_{n_i}(Z_i;\theta)$$ where the $$Z_1,...,Z_{n_c}$$ are mutually independent group level data such that $$E(\varphi_{n_i}(Z_i;\theta))=0$$ for all $$i=1,...n_c$$ and $$n_i$$ is cluster size. Suppose solving the unbiased estimating equations

$$\sum_{i=1}^{n_c} \varphi_{n_i}(Z_i;\theta) = 0$$

gives you the GEE estimators. For simplicity sake, assume the variance of the random effects is known. The $$\varphi_{n_i}(Z_i;\theta)$$ are independent and of the same dimension BUT they are only mutually independent; indeed given $$n_i$$ is not constant, they will in general not be identically distributed. The proof of asymptotic normality of the coefficients requires (among other things) that $$\sqrt{n_c}\left (\frac{1}{n_c} \sum_{i=1}^{n_c} \varphi_{n_i}(Z_i;\theta)\right )$$ converges in distribution to a mean zero multivariate normal. What version of CLT is typically appealed to to make this statement?

The application to vectors comes from the Cramér-Wold theorem that a sequence of vectors $$Z_n$$ converges to a Normal vector $$Z$$ if and only if $$b'Z_n\stackrel{d}{\to} b'Z$$ for every unit vector $$b$$.
Since the Lyapunov CLT conditions are in terms of moments, they translate conveniently: the univariate sequence $$b'Z_n$$ have bounded third moments for every $$b$$ if and only if each element of the vector $$Z_n$$ has bounded third moments. For your $$\varphi_{n_i}$$ that's going to follow from a moment condition on the individual-observation contributions to $$\varphi$$ and a uniform bound on the sizes $$n_i$$ of clusters.