Page 130 - Elements of Distribution Theory

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116 Moments and Cumulants

Although cumulants may be calculated directly from the characteristic function, there
are relatively few cases in which the moment-generating function of a distribution does not
exist, while the characteristic function is easily calculated and easy to differentiate. It is
often simpler to calculate the cumulants by calculating the moments of the distribution and
then using the relationship between cumulants and moments.

Example 4.20 (Log-normal distribution). Let X denote a random variable with the
log-normal distribution considered in Example 4.14. Recall that, although the moment-
generating function of this distribution does not exist, all moments do exist and are
given by

1 2
r
E(X ) = exp r , r = 1, 2,....
2
Hence,
κ 1 = exp(1/2),κ 2 = exp(2) − exp(1),κ 3 = exp(9/2) − 3exp(5/2) + 2exp(3/2),

and so on.

Earlier in this section, the relationship between the the cumulants of a linear function
of a random variable and the cumulants of the random variable itself was described. The
following theorem gives a formal statement of this relationship for the more general case
in which the moment-generating function does not necessarily exist.

Theorem 4.14. Let X denote a real-valued random variable with mth cumulant κ m (X) for
some m = 1, 2,... and let Y = aX + b for some constants a, b. Then the mth cumulant of
Y, denoted by κ m (Y),is given by

aκ 1 (X) + b if m = 1
κ m (Y) = m .
a κ m (X) if m = 2, 3,...
Proof. Let ϕ X and ϕ Y denote the characteristic functions of X and Y, respectively. We
have seen that ϕ Y (t) = exp(ibt)ϕ X (at). Hence,

log(ϕ Y (t)) = ibt + log(ϕ X (at)).
m
If E(|X| ) < ∞ then, by Theorem 4.13,
m
j m
log(ϕ X (t)) = (it) κ j (X)/j! + o(t )as t → 0;
j=1
it follows that
m
j m
log(ϕ Y (t)) = ibt + (ita) κ j (X)/j! + o(t )
j=1
m
j j m
= (it)(aκ 1 (X) + b) + (it) a κ j (X)/j! + o(t ).
j=2
The result now follows from Theorem 4.13.