Let T n be a random variable having a non-central chi-squared distribution with n degrees of freedom and non-centrality parameter   Show T n as the same asymptotic distribution as X 1 +···+ X n , for...

Let T_n
be a random variable having a non-central chi-squared distribution with n degrees of freedom and non-centrality parameter
  Show T_n
as the same asymptotic distribution as X₁
+···+ X_n, for all n ≥ 1, where the Xj are i.i.d. random variables following a noncentral chi-squared distribution with 1. degree of freedom and non-centrality parameter
  Hence, use and show that
  is asymptotically normal. In particular, for
  verify that