Let X1,,Xn be independent and identically distributed scalar random variables having cmnulants 1, 2, 3, . Show

Let X1,…,Xn be independent and identically distributed scalar random variables having cmnulants κ1, κ2, κ3, …. Show that where i2

= −1. Hence, by writing ω = e2πi/r and rκr = limn→∞

n

−1[X1 + ωX2 + ω

2X3 + … + ω

nr−1Xnr]

r

, give an interpretation of cumulants as coefficients in the Fourier transform of the randomly ordered sequence X1,X2,…. Express κr as a symmetric function of Χ1,Χ2,…, (Good, 1975, 1977).