Page 128 - Elements of Distribution Theory

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114 Moments and Cumulants

Hence, by Leibnitz’s rule,
r
d r d j d r− j
r

α (t) = β (t) α(t).
dt r j dt j dt r− j
j=0
The result now follows by evaluating both sides of this expression at t = 0.
Hence, applying this result to the moment-generating and cumulant-generating functions
yields a general formula relating moments to cumulants. In this context, α 1 ,α 2 ,... are the
moments and β 1 ,β 2 ,... are the cumulants. It follows that
r
r r− j
r+1
E(X ) = κ j+1 E(X ), r = 0, 1,....
j
j=0
r
An important consequence of this result is that κ r is a function of E(X),..., E(X ).
Lemma 4.1 can also be used to derive an expression for central moments in terms of
cumulants by interpreting α 1 ,α 2 ,... as the central moments and β 1 ,β 2 ,..., under the
assumption that α 1 = β 1 = 0. Hence,
2 3
E[(X − µ) ] = κ 2 , E[(X − µ) ] = κ 3
and
2
4
E[(X − µ) ] = κ 4 + 3κ .
2
The approach to cumulants taken thus far in this section requires the existence of the
moment-generating function of X.A more general approach may be based on the charac-
m
teristic function. Suppose that X has characteristic function ϕ X (t) and that E(X )exists and
(m) (m)
is finite. Then, by Theorem 3.5, ϕ (0) exists and, hence, the mth derivative of log ϕ (t)
X X
at t = 0exists. We may define the jth cumulant of X,1 ≤ j ≤ m,by
1 d j

κ j = log ϕ X (t) .
j
(i) dt j t=0
Of course, if the cumulant-generating function X exists, it is important to confirm that
the definition of cumulants based on the characteristic function agrees with the definition
based on the cumulant-generating function. This fact is established by the following lemma.
Lemma 4.2. Let X denote a random variable with moment-generating function M X and
characteristic function ϕ X . Then, for any m = 1, 2,...,
d m 1 d m

= log ϕ X (t) .
m
log M X (t)
dt m t=0 (i) dt m t=0
Proof. Fix m. Since the mth moment of X exists we may write
m
j j m
M X (t) = 1 + t E(X )/j! + o(t )as t → 0
j=1
and
m
j j m
ϕ X (t) = 1 + (it) E(X )/j! + o(t )as t → 0.
j=1