Page 120 - Elements of Distribution Theory

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106 Moments and Cumulants

Example 4.14 (Log-normal distribution). Let X denote a random variable with the log-
normal distribution considered in Example 4.12; recall that the moment-generating function
of this distribution does not exist. Note that
∞

r r 1 1 2
E(X ) = x √ exp − [log(x)] dx
x (2π) 2
0
∞ 1 1

= exp(rt)√ exp − t 2 dt
(2π) 2
−∞

1 2
= exp r .
2
Hence, all the moments of the distribution exist and are finite, even though the moment-
generating function does not exist. That is, the converse to the first part of Theorem 4.8
does not hold.
If two moment-generating functions agree in a neighborhood of 0, then they represent
the same distribution; a formal statement of this result is given in the following theorem.

Theorem 4.9. Let X and Y denote real-valued random variables with moment-generating
functions M X (t), |t| <δ X , and M Y (t), |t| <δ Y ,respectively. X and Y have the same dis-
tribution if and only if there exists a δ> 0 such that
M X (t) = M Y (t), |t| <δ.

Proof. Note that, since M X (t) and M Y (t) agree in a neighborhood of 0,
j
j
E(X ) = E(Y ), j = 1, 2,....
Since, for |t| <δ,
E(exp{t|X|}) ≤ E(exp{tX}) + E(exp{−tX}) < ∞,

it follows that moment-generating function of |X| exists and, hence, all moments of |X|
exist. Let

j
γ j = E(|X| ), j = 1, 2,....
Since

∞
j
γ j t /j! < ∞ for |t| <δ,
j=0
it follows that
γ j t j
lim = 0, |t| <δ.
j→∞ j!
By Lemma A2.1,

n n+1

|hx|
j

exp{ihx}− (ihx) /j! ≤ .
(n + 1)!
j=0