## Statistical "Error" of Centroid of Gaussian Distribution

If I have L data samples, distributed randomly (3D real Gaussian distibution, unity variance) about the origin in 3D real space, how can I derive an expression for the "origin estimation error" (i.e. the distance between the true origin and the centroid of the data points) as a function of L?

Intuitively, as L->infinity, the error->0. In fact, it is easy to show in Matlab that the error falls as 1/sqrt(L) (for sufficiently large number of trials). However, I don't know where to start with a proof. (I'm really trying to write a proof for N-dim complex space, but I expect that will only need an extra sqrt(2) term).