Let X1,,Xn be independent and identically distributed p-dimensional random vectors having cumulants r , r,s

Let X1,…,Xn be independent and identically distributed p-dimensional random vectors having cumulants κ

r

, κ

r,s

, κ

r,s,t

,…. Define the random vector Z(n)

by Z

r

(n) =

n∑

j=1 Xr j exp (2πij/n)

where i2 = −1. Using the result in the previous exercise or otherwise, show that the nthorder moments of Z(n)

are the same as the nth-order cumulants of X, i.e.

E (Z r1

(n) …Z rn

(n)) = κ

r1,…,rn

,

(Good, 1975, 1977). Hence give an interpretation of mixed cumulants as Fourier coefficients along the lines of the interpretation in Exercise 2.25.