Looking for a law of large numbers

Question

I am not familiar with laws of large numbers (LLN) and I have a question on whether some LLN is applicable to the following setting:

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Assumption (*): Let $i\in I\equiv \{1,...,n\}$ and $j\in J\equiv \{1,...,m\}$. Consider the following random variables defined on a probability space $(\Omega, \mathcal{F}, P)$: $X_{ij}, Z_{ij}, X_i, Z_j$ $\forall i\in I$, $\forall j \in J$ (for a total of $2nm+n+m$ random variables).

Assume $\Big(X_{ij}, Z_{ij}, X_i, Z_j\text{ }\forall i\in I \text{ } \forall j \in J\Big)$ are i.i.d. with standard normal cdf.

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$\forall i\in I$, $\forall j \in J$, define the random variable $$ W_{ij}\equiv X_{ij}+Z_{ij}+X_{i}+Z_{j} $$ By Assumption (*), $W_{ij}\sim N(0,4)$ and we denote its cdf by $\Phi$.

We can see that $\Big(W_{ij}\text{ }\forall i\in I \text{ } \forall j \in J\Big)$ are identically distributed but not mutually independent (e.g., $W_{11}=X_{11}+Z_{11}+X_{1}+Z_{1}$ and $W_{12}=X_{12}+Z_{12}+X_{1}+Z_{2}$ which are clearly not independent).

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Define $\hat{F}_{nm}(x)=\frac{1}{nm}\sum_{i,j\in I\times J}1\{W_{ij}\leq x\}$ for any $x\in \mathbb{R}$.

Since $\Big(W_{ij}\text{ }\forall i\in I \text{ } \forall j \in J\Big)$ are not i.i.d., we cannot use the strong LLN to say that $\hat{F}_{nm}(x)\rightarrow_{a.s.}\Phi(x)$ as $nm\rightarrow \infty$.

Still, does Assumption (*) tell us something about convergence (almost surely or in probability) of $\hat{F}_{nm}(x)$ as $nm\rightarrow \infty$?

Answer 1

A spot-on paper appears to be

Azriel, D., & Schwartzman, A. (2015). The empirical distribution of a large number of correlated normal variables. Journal of the American Statistical Association, 110(511), 1217-1228.

The paper's abstract:

"Motivated by the advent of high-dimensional, highly correlated data, this work studies the limit behavior of the empirical cumulative distribution function (ecdf) of standard normal random variables under arbitrary correlation. First, we provide a necessary and sufficient condition for convergence of the ecdf to the standard normal distribution. Next, under general correlation, we show that the ecdf limit is a random, possible infinite, mixture of normal distribution functions that depends on a number of latent variables and can serve as an asymptotic approximation to the ecdf in high dimensions. We provide conditions under which the dimension of the ecdf limit, defined as the smallest number of effective latent variables, is finite. Estimates of the latent variables are provided and their consistency proved. We demonstrate these methods in a real high-dimensional data example from brain imaging where it is shown that, while the study exhibits apparently strongly significant results, they can be entirely explained by correlation, as captured by the asymptotic approximation developed here. Supplementary materials for this article are available online."