Your double subscripts are extraneous. Let's consider a simpler situation, where we have a single family of random variables $\{X_i\}$.

As Yuri Bakhtin says above, your condition is not sufficient for a CLT to hold. Here is a simpler situation, however: suppose that $X_i$ and $X_j$ satisfy finite-range dependence. That is, there exists a positive integer $R$ such that if $|i-j| \ge R$, then $X_i$ and $X_j$ are independent. We will prove a law of large numbers for $\{X_i\}$. If you're interested, you can push it farther to prove a central limit theorem. Suppose that $X_i$ has mean $\mu$ for each $i$.

Let $S_N = \tfrac{1}{N} \sum_{i=1}^N X_i$ as usual. Without loss of generality, we may consider indices only divisible by $R$: $S_{RN} = \tfrac{1}{RN} \sum_{i=1}^{RN} X_i$. Let $$S_{RN}^{(k)} = \tfrac{1}{N} \sum_{j=0}^{N-1} X_{Rj+k}$$ for $k= 1, \dots, R$, so that $$S_{RN} = \tfrac{1}{R} \left( S_{RN}^{(1)} + \dots + S_{RN}^{(R)} \right).$$Each sum $S_{RN}^{(k)}$ is comprised of independent random variables, so the classical law of large numbers applies and $S_{RN}^{(k)} \to \mu$ both in probability and almost surely. Consequently, $S_{RN} \to \mu$.

Obviously, this argument breaks down when $R = \infty$. In that case, the problem is no longer trivial and you will have to be more cautious with your assumptions.