Let Y = Y 1 ,,Y p have zero mean and covariance matrix i,j . Show

Let Y = Y 1

,…,Y p

have zero mean and covariance matrix κ

i,j

. Show that the

‘total variance’, σ

2 = E(Y iY jδij

), is invariant under orthonormal transformation of Y. For any given direction, ϵ, define Q4 = a0 + aiXi + aijXiXj/2! + aijkXiXjXk/3!

+aijklXiXjXkXl/4!,

κ2 (U) = ∑

1≤i≤j≤n

β

2j−2i−2

= (1 − β

2)

−1 {n − (1 − β

2n)/ (1 − β

2)} = E (T2)

κ3 (U) = 6 ∑

1≤i≤j≤k≤n

β

2k−2i−3

=

6{