Consider $ R = \sum_{j=1}^{N} a_{j}b $ where b is a bernoulli random variable with $Probability(b=1) = p$ and $a_{j} \in \mathbb{R}^2$ is a vector . There is a line $Y = mX$ such that $ \sum_{a_{j}(2)>ma_{j}(1)} a_{j} = \sum_{a_{j}(2)<ma_{j}(1)} a_{j}$ i.e. the points $a_{j}$ are symmetrically distributed about this axis. In other words, the centroid of all points $a_{j}$ lies on this axis.
I calculated the mean and variance of R using standard techniques and got them to be
$E(R) = pC$ where $C$ is the centroid of all $a_{j}$. and
$ V(R) = p(1-p)(-(N+1)CC^{T} + \frac {N^2}{(N-1)^2} \sum_{j=1}^{N} a_{j}a_{j}^T) $
For large N the distribution will tend to a Normal distribution owing to the central limit theorem but I feel that owing to the symmetry of the weights, the distribution is "Gaussian like". How do I go about proving this, without having to rely on the large N assumption? I would really like to show that the R acts like a random variable with normal distribution.