Page 132 - Elements of Distribution Theory

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118 Moments and Cumulants

The standardized cumulants satisfy

( j−1)
ρ j (S) = ρ j /n 2 , j = 3, 4,...
where ρ 3 ,ρ 4 ,... denote the standardized cumulants of X 1 .

Cumulants of a random vector
Let X = (X 1 ,..., X d ) denote a d-dimensional random vector. Joint cumulants of elements
of X may be defined using the same approach used to define the cumulants of a real-valued
random variable. For simplicity, we consider only the case in which the moment-generating
function of X exists. However, as in the case of a real-valued random variable, the same
results may be obtained provided only that moments of a certain order exist.
Let M denote the moment-generating function of X with radius of convergence δ> 0.
Then the cumulant-generating function of X is given by K(t) = log M(t) and the joint
cumulant of order (i 1 ,..., i d ), where the i j are nonnegative integers, is given by

∂ i 1 + ··· +i d
= K(t) ;
κ i 1 ··· i d t=0
i 1
∂t ··· ∂t i d
1 d
here t = (t 1 ,..., t d ). Although this definition may be used to define joint cumulants of
arbitrary order, the most commonly used joint cumulants are those in which i 1 + ··· +
i d = 2, for example, κ 110···0 , κ 1010···0 , and so on.
The following result gives some basic properties of joint cumulants.

Theorem 4.16. Let X = (X 1 ,..., X d ) denote a d-dimensional random vector with
cumulant-generating function K.
(i) Fix 1 ≤ j ≤ d and assume that

i k = 0.
k
= j
Then the joint cumulant of order (i 1 ,..., i d ) is the i j th cumulant of X j .
(ii) Suppose that, for some 1 ≤ j < k ≤ d, i j = i k = 1 and i 1 +· · · + i d = 2. Then the
.
joint cumulant of order (i 1 ,..., i d ) is the covariance of X i j and X i k
(iii) Suppose that, for 1 ≤ j < k ≤ d, X j and X k are independent. Then any joint cumu-
lant of order (i 1 ,..., i d ) where i j > 0 and i k > 0 is 0.
(iv) Let Y denote a d-dimensional random variable such that all cumulants of Y exist and
(X + Y)
assume that X and Y are independent. Let κ i 1 ···i d (X), κ i 1 ···i d (Y), and κ i 1 ···i d
denote the cumulant of order (i 1 ,..., i d ) of X, Y, and X + Y, respectively. Then
(Y).
κ i 1 ···i d (X + Y) = κ i 1 ···i d (X) + κ i 1 ···i d

Proof. Consider part (i); without loss of generality we may assume that j = 1. Let K 1
denote the cumulant-generating function of X 1 . Then

K 1 (t 1 ) = K((t 1 , 0,..., 0)).

Part (i) of the theorem now follows.