Spherically symmetric random variables: A random variable X is said to be spherically symmetric if its distribution is unaffected by orthogonal transformation.

Equivalently, in spherical polar coordinates, the radial vector is distributed independently of the angular displacement. Show that the odd cumulants of such a random variable are zero and that the even cumulants must have the form

κ

i,j = τ2δ

ij

, κ

i,j,k,l = τ4δ

ijδ

kl

[3], κ

i,j,k,l,m,n = τ6δ

ijδ

klδmn

[15]

and so on, for some set of coefficients τ2, τ4, …. Show that the standardized cumulants are

ρ4 = τ4 (p + 2)/τ

2 2

, ρ6 = τ6 (p + 2)(p + 4)/τ

3 2

and hence that τ4 ≥ −2τ

2 2 / (p + 2).