What is the Cumulative Distribution Function of the following random variable? 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 $P((k∑_iX_iY_i+(∑_iX_i)(∑_jY_j))<c)$ for any integer c.
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? 

$$Z = \sum_{i=1}^n \sum_{j=1}^n X_i Y_j = \left(\sum_{i=1}^n X_i\right)\left(\sum_{j=1}^n Y_j\right)$$
If $n$ is large, $S_X = \sum_i X_i$ and $S_Y = \sum_j Y_j$ are approximately normal. They have means $n\mu$ and standard deviations $\sqrt{n} \sigma$ where each $X_i$ and $Y_j$ have mean $\mu$ and standard deviation $\sigma$.  Of course they are independent.  Thus
$E[Z] = E[S_X] E[S_Y] = n^2 \mu^2$ and $E[Z^2] = E[S_X^2] E[S_Y^2] = (n^2 \mu^2 + n \sigma^2)^2$, so the variance of $Z$ is $\text{Var}(Z) = E[Z^2] - E[Z]^2 = n^2 \sigma^4 + 2 n^3 \sigma^2 \mu^2$.  
The moment generating function of the product of independent normal random variables with means $n\mu$ and standard deviations $n \sqrt{\sigma}$ has, according to Maple, moment generating function
$$ M_Z(t) = E[e^{tZ}] = \frac{1}{\sqrt{1 - n^2 \sigma^4 t^2}} \exp\left(\frac{n^2 \mu^2 t}{1 - n \sigma^2 t}\right)$$
for  $t < 1/(n \sigma^2)$.
EDIT: If $\mu \ne 0$, it would be better to separate out the effect of the mean.  So let $X_i = \mu + \sigma U_i$ and $Y_i = \mu + \sigma V_i$, where $U_i$ and $V_i$ have mean $0$ and standard deviation $1$.  Then $$Z  = n^2 \mu^2 + n \mu \sigma \sum_{i=1}^n (U_i + V_i) + \sigma^2 \sum_{i=1}^n \sum_{j=1}^n U_i V_j$$
Now $n \mu \sigma \sum_{i=1}^n (U_i + V_i)$ is approximately normal with mean $0$ and 
standard deviation $\sqrt{2} n^{3/2} \mu \sigma$, while $\sigma^2 \sum_{i=1}^n \sum_{j=1}^n U_i V_j$ has mean $0$ and standard deviation $n \sigma^2$.  For large
$n$ this term is negligible compared to the $n^{3/2}$ term.  So a good
approximation to the distribution of $Z$ is normal with mean $n^2 \mu^2$ and standard deviation $\sqrt{2} n^{3/2} \mu \sigma$.
You asked about $ (k−1) \sum_i X_i Y_i+ Z$: call this $(k-1) T + Z$.  If we separate out the effect of the mean, 
$$T = n \mu^2 +  \mu \sigma \sum_{i=1}^n (U_i + V_i) + \sigma^2\sum_{i=1}^n U_i V_i$$
where $\mu \sigma \sum_{i=1}^n (U_i + V_i)$ has mean $0$ and standard deviation $\sqrt{2n} \mu \sigma$ and $\sigma^2 \sum_{i=1}^n U_i V_i$ has mean $0$ and standard deviation $\sqrt{n} \sigma^2$.  Again, these terms are negligible compared to the $n^2$ and $n^{3/2}$ terms.  
