Show that Y r defined in the previous exercise has third cumulant of order O(n−3/2

) and fourth cumulant of order O(n−1

). Hence show that 2nK*(X̄n

) has a non-central X 2 p

distribution for which the rth cumulant is

{1 + b/n}

r2 r−1 (r − 1)!p + O (n

−2).

Find an expression for b in terms of the invariant cumulants of X.