pr.probability - Does central limit theorem hold for general weakly dependent variables?

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.