I am looking for a LLN applicable to the following setting:
Let $\mathcal{N}\equiv \{1,...,n\}$. Let $R: \mathcal{N}\rightarrow \mathcal{N}$ increasing on $\mathcal{N}$ (e.g., $R(n)=n$)
Assumption (*):
$\{X_{ij},Z_{ij} \text{ }\forall i\in \mathcal{N}, \forall j\in \mathcal{N}\}$ are r.v. identically distributed as Gumbel with location $0$ and scale $1$.
$Q_{i}\equiv \max \{Q_{i,1},..., Q_{i,R(n)}\}$ $\forall i \in \mathcal{N}$ and $T_{j}\equiv \max \{T_{j,1},..., T_{j,R(n)}\}$ $\forall j \in \mathcal{N}$, where $\{T_{i,k}, T_{j,k} \text{ }\forall i\in \mathcal{N}, \forall j\in \mathcal{N} ,\forall k \in \{1,...,R(n)\}\}$ are r.v. identically distributed as Gumbel with location $0$ and scale $1$
$\{X_{ij}, Z_{ij}, Q_{i,k}, T_{j,k} \text{ }\forall i\in \mathcal{N}, \forall j\in \mathcal{N} ,\forall k \in \{1,...,R(n)\}\}$ are mutually independent
$\forall i\in \mathcal{N}$, $\forall j \in \mathcal{N}$, define the random variable $$ W_{ij}\equiv -X_{ij}-Z_{ij}+Q_{i}+T_{j} $$ Under Assumption (*), $W_{ij}$ is distributed approximatively as a logistic with location $2\log(R(n))-2$ and scale $2$ and we denote its cdf by $F_n$.
Sketch of the proof: $Q_i$ is distributed as a Gumbel with scale $1$ and location $\log(R(n))$. Similarly $T_j$. $Q_i-X_{ij}$ is distributed as a Logistic with scale $1$ and location $\log(R(n))-1$. Similarly $T_j-Z_{ij}$. The sum of two logistic is approximatively a logistic with scale and location given by the sum of scale and location of the two Logistic.
We can see that $\{W_{ij}\text{ }\forall i\in \mathcal{N} \text{ } \forall j \in \mathcal{N}\}$ are identically distributed but not mutually independent (e.g., $W_{11}=X_{11}+Z_{11}+Q_{1}+T_{1}$ and $W_{12}=X_{12}+Z_{12}+Q_{1}+T_{2}$ which are clearly not independent).
Define $\hat{F}_{n}(x)\equiv\frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n1\{W_{ij}\leq x\}$ for any $x\in \mathbb{R}$, where $1$ is the indicator function equal to $1$ if the condition inside is satisfied and $0$ otherwise.
If $\{W_{ij}\text{ }\forall i\in \mathcal{N} \text{ } \forall j \in \mathcal{N}\}$ were mutually independent, we could state
$$ \sup_{x\in \mathbb{R}}|\hat{F}_{n}(x)-F_n(x)|\rightarrow_{a.s.} 0 \text{ as $n\rightarrow \infty$} $$
Question: Still, does Assumption (*) allow us to state convergence (almost surely or in probability, uniform or pointwise) of $\hat{F}_{n}(x)-G_n(x)$ to $0$ as $n\rightarrow \infty$ for some $G_n$? Simulations reveal that in any case $\hat{F}_{n}(x)-F_n(x)$ does not converge in any sense to $0$.