Show that ij ( ij) 2 = n 2/ (n 1). Hence derive the following joint

Show that

∑ij

(ϕ

ij)

2 = n 2/ (n − 1).

Hence derive the following joint cumulants of k-statistics:

Show explicitly that the term κ

r,t,uκ

s,v does not appear in the third cumulant, κk

(r∣s∣t,u,v).