Suppose that we have $2n$ iid random variables $X_1,…,X_n,Y_1,…,Y_n$ where $n$ is a large number. I want to find $P((k∑_iX_iY_i+(∑_iX_i)(∑_jY_j))
Since $n$ is a large number and all the random variables are $iid$, using central limit theorem, we can say that $k∑_iX_iY_i$, $(∑_iX_i)$ and $(∑_jY_j)$ are approximately normal random variables and $(∑_iX_i)$$(∑_jY_j)$ is the product of two normal random variables which would have Normal Product Distribution.
So $k∑_iX_iY_i+(∑_iX_i)(∑_jY_j)$ is the sum of one normal and one normal product random variable which are dependent.
Now the question is how can we find $P((k∑_iX_iY_i+(∑_iX_i)(∑_jY_j)) \le c)$ for any integer c?