Fix a number m and let us take a set, say A, of unit vectors {v_1,...,v_k} in R^n. Assume that k is large, say exponentially large in n (k=e^{cn}). Let X be the euclidean length of a random sum of m vectors in A (all sums are equally likely).

It is quite intuitive that the typical length of such sum should be sqrt(m) and it is not that hard to verify it.

Question: How does the variance of X behave? More precisely, is it true that VarX=o(sqrt(m)). If so, could one get a better rate? Maybe it is m^1/4?

I would be very grateful for any information or link.