Originally Posted by

**akbar** Using this result, in the case of the 3d random walk, the correlation matrix of the increment vector is diagonal (this is immediate from the probability mass function). E.g. the vector components are uncorrelated (yet not independent).

This ensures that the component of the limiting Gaussian vector are independent (2 by 2), since in the case of multivariate Gaussian variables it is implied by the fact they are uncorrelated. This enables to move to the independence case and apply your solution as it is with the limiting distribution.