Show that there exists a unique distribution whose odd cumulants are zero and whose even cumulants are κ2 = 1, κ4 = −2, κ6 = 16, κ8 = −272, κ10 =

7936,…. Let M′r

(λ) be the moment matrix described in Section 3.8, corresponding to the cumulant sequence, λκ1, …, λκ2r

. Show that for the particular cumulant sequence above, the determinant of M′r

(λ) is

|M ′

r| = 1!2!…r!λ

r(λ − 1)

r−1 …(λ − r + 1), for r = 1, 2, 3, 4. Hence prove that there is no distribution whose cumulants are

{λκr} for non-integer λ < 3. Find the unique distribution whose cumulants are

{λκr} for λ = 1, 2, 3.