Page 94 - Elements of Distribution Theory

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052184472Xc03 CUNY148/Severini May 24, 2005 2:34

80 Characteristic Functions

The remainder of the proof is essentially the same as the m = 1 case considered above.
Since
[i sin(hX/2)]
2m
ϕ (2m) (0) = lim E X 2m
h→0 (hX/2) 2m
and
2m
sin(hX/2) 2m
(2m)
|ϕ (0)|= lim E X ,
h→0 (hX/2) 2m
and by Fatou’s Lemma (see Appendix 1),
sin(hX/2) 2m 2m
2m
lim E X ≥ E(X ),
h→0 (hX/2) 2m
it follows that

E(X 2m ) ≤|ϕ (2m) (0)|
so that E(X 2m ) < ∞.

Example 3.9 (Normal distribution). Let X denote a random variable with a normal dis-
tribution with parameters µ and σ; see Example 3.6. This distribution has characteristic
function
σ 2
2
ϕ(t) = exp − t + iµt .
2
r
Clearly, ϕ(t)is m-times differentiable for any m = 1, 2,... so that E(X )exists for all
r = 1, 2,.... It is straightforward to show that
2
2

ϕ (0) = µ and ϕ (0) =−(µ + σ )
so that
2
2
2
E(X) = µ and E(X ) = µ + σ .
Example 3.10 (Gamma distribution). Consider a gamma distribution with parameters α
and β,asin Example 3.4; recall that this distribution has characteristic function
β α
ϕ(t) = , t ∈ R.
(β − it) α
Since the density function p(x) decreases exponentially fast as x →∞,itis straightforward
m
to show that E(X )exists for any m and, hence, the moments may be determined by
differentiating ϕ.
Note that
β α
(m) m
ϕ (t) =−(−i) [α(α + 1) ··· (α + m − 1)]
(β − it) α+m
so that
1
(m)
m
ϕ (0) =−(−i) [α(α + 1) ··· (α + m − 1)]
β m