Let X be a scalar random variable and write M (n) X () = 1 +

Let X be a scalar random variable and write M

(n)

X

(ξ) = 1 + μ

′

1ξ + μ

′

2ξ

2/2! + … + μ

′

nξ

n/n!

for the truncated moment generating function. The zeros of this function, a

−1 1

,…, a−1 n

, not necessarily real, are defined by M

(n)

X

(ξ) = (1 − a1ξ)(1 − a2ξ)…(1 − anξ).

Show that the symmetric functions of the as pρ

2 13 = E (R 3

1R 3

2 cos θ12)

pρ

2 23 = E (R3 1R3 2cos 3

¯¯¯¯θ12)

⟨rs…u⟩ = ∑

i

r ia s

j …a u

k are semi-invariants of X (unaffected by the transformation X → X +

c) if and only if the powers r,s,…,u that appear in the symmetric function are at least 2. Show also that the cumulants are given by the particular symmetric functions

.

Express the semi-invariant (22) in terms of κ2 and κ4. (MacMahon, 1884, 1886; Cayley, 1885).