For given 0 < r < n, the real Grassmanian Gr(r,n) is a specific compact subset of the boundary at infinity of the symmetric space X of SL(n,R), where X is identified with the real, symmetric and positive definite n x n matrices of determinant one. Fix any Borel probability measure p on Gr(r,n) and let U_k be random samples with values in Gr(r,n) that are i.i.d. accordingly to p. Under some assumptions, with p there is associated a unique matrix M in X. Similarly, with each empirical probability measure p_k on Gr(r,n) corresponding to the samples U_1, …, U_k there is associated a unique random matrix M_k in X. Then a generalization of Law of Large Numbers holds true: M_k converges to M almost surely. As well, we also provide a Central Limit Theorem for a rescaled process of M_k. Those results are proven by making use of the specific geometry related to the symmetric space X and the explicite formulas of gradients and Hessians. This is a joint work with Christian Mazza.