What is the equivalent smoothing function to running the same Gaussian 8 times? Suppose there is an image which is to be smoothed by convolving it with a Gaussian kernel with standard deviation $\sigma$. If the image is then smoothed with this kernel 8 consecutive times, is the total smoothing still Gaussian with some calculable standard deviation $\hat{\sigma}$? Or is it some other distribution instead?
From Wikipedia:


  Applying multiple, successive gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied. 


So in your case, that would be equivalent to blurring the image with a single Gaussian with $\hat{\sigma}$:
$$
\hat{\sigma} = \sqrt{8\sigma^2} = 2\sqrt{2} \sigma
$$