Let $A_{j,n}$ and $B_{j,n}$ be two triangular arrays of independent Bernoulli variables such that
$$ \begin{array}{l} \mathbb{P}\left[A_{j,n}=1\right] = p_{A,n},~~\mathbb{P}\left[A_{j,n}=0\right] = 1-p_{A,n},\\ \mathbb{P}\left[B_{j,n}=1\right] = p_{B,n},~~\mathbb{P}\left[B_{j,n}=0\right] = 1-p_{A,n}. \end{array} $$
Besides when $n\rightarrow\infty$ we have $p_{A,n}\rightarrow p_A\in(0,1)$ and $p_{B,n}\rightarrow p_B\in(0,1)$. Let know $\Gamma_n$, $\gamma_{A,n}$ and $\gamma_{B,n}$ be the random variables defined as
$$ \begin{array}{l} \Gamma_n = \frac{1}{n}\,\sum_{j=1}^nA_{j,n}\cdot B_{j,n},\\ \gamma_{A,n} = \frac{1}{n}\,\sum_{j=1}^nA_{j,n},\\ \gamma_{B,n} = \frac{1}{n}\,\sum_{j=1}^nB_{j,n}. \end{array} $$
Obviously the product $A_{j,n}\cdot B_{j,n}$ is still a triangular array of Bernoulli variables, with new probability of begin $1$ or $0$. Define now
$$ \Phi_n = \frac{\Gamma_n-\gamma_{A,n}\cdot \gamma_{B,n}}{1+\Gamma_n-\gamma_{A,n}-\gamma_{B,n}}. $$
I need to find the asymptotic distribution of $\Phi_n$ when $n\rightarrow\infty$.
For the variable $\Gamma_n$, $\gamma_{A,n}$ and $\gamma_{B,n}$ alone I can use a standard CTL theorem, since they are the average of iid variables. Nevertheless I can't still find the distribution of $\Phi_n$ since it involves a non-trivial combination of the variables. I am pretty convinced that the continuous mapping theorem should be somehow invoked.