Explicit Expressions for the first 5 moments of Spearman's RHO and a partial(elementary!) proof of its  Asymptotic Normality



                              By Shalosh B. Ekhad



Let's define the following "measure of disarray" on permutations of length n, let's call it RHO(pi)



                                      n
                                    -----
                                     \               2
                          RHO(pi) =   )   (pi[i] - i)
                                     /
                                    -----
                                    i = 1



                           The expectation of RHO is



                               n (n - 1) (n + 1)
                               -----------------
                                       6



                             and in Maple notation



1/6*n*(n-1)*(n+1)




                             The variance of RHO is



                               2                2
                              n  (n - 1) (n + 1)
                              -------------------
                                      36



                             and in Maple notation



1/36*n^2*(n-1)*(n+1)^2


                  The, 3, -th moment about the mean of RHO is



                                       0



                             and in Maple notation



0


                  The, 4, -th moment about the mean of RHO is



                 3              3       2                     3
                n  (n - 1) (25 n  - 38 n  - 35 n + 72) (n + 1)
                -----------------------------------------------
                                     10800



                             and in Maple notation



1/10800*n^3*(n-1)*(25*n^3-38*n^2-35*n+72)*(n+1)^3


                  The, 5, -th moment about the mean of RHO is



                                       0



                             and in Maple notation



0


                      Let's now look at the scaled moment



        The limit of the, 3, -th scaled moment about the mean of RHO is



0


        The limit of the, 4, -th scaled moment about the mean of RHO is



3


                       more precisely, the asymptotic is



            114     6      102     6      102     6      102       1
        3 - ---- - ---- + ----- - ---- + ----- - ---- + ----- + O(----)
            25 n      2       3      4       5      6       7       8
                   5 n    25 n    5 n    25 n    5 n    25 n       n



                             and in Maple notation



3-114/25/n-6/5/n^2+102/25/n^3-6/5/n^4+102/25/n^5-6/5/n^6+102/25/n^7+O(1/n^8)
        The limit of the, 5, -th scaled moment about the mean of RHO is



0


This proves asymptotic normality up to the 5th moment in a fully elementary \
    way, confirming the fact that RHO is, most probably, asymptotically norm\
    al