Page 104 - Elements of Distribution Theory

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

90 Characteristic Functions

Let

t = inf{t ∈ R: t > 0 and |ϕ(t)|= 1}.
∗
∗
Then 2π/t is the maximal span of the distribution and
|ϕ(t)| < 1 for |t| < t .
∗
That is, the maximal span b = 2π/t satisfies
∗
|ϕ(2π/b)|= 1
and

|ϕ(t)| < 1 for |t| < 2π/b,
proving part (ii).
Let p j = Pr(X = a + bj), j = 0, ±1, ±2,.... Then
∞

ϕ(t) = p j exp{it(a + bj)}
j=−∞
and
∞

ϕ(t + 2πk/b) = p j exp{it(a + bj)} exp{i2πk(a + bj)/b}
j=−∞
∞

= exp(i2πka/b) p j exp{it(a + bj)} exp(i2πkj).
j=−∞
Part (iii) of the theorem now follows from the fact that

exp(i2πkj) = cos(kj2π) + i sin(kj2π) = 1.

Example 3.19 (Binomial distribution). Consider a binomial distribution with parameters
n and θ. The characteristic function of this distribution is given by

ϕ(t) = [1 − θ + θ exp(it)] n
so that
n
2
2
|ϕ(t)|= [(1 − θ) + 2θ(1 − θ) cos(t) + θ ] 2 .
Hence, |ϕ(t)|= 1if and only if cos(t) = 1; that is, |ϕ(t)|= 1if and only if t = 2π j for
some integer j.
Thus, according to part (1) of Theorem 3.11, the binomial distribution is a lattice distri-
bution. According to part (ii) of the theorem, the maximal span is 1.

3.4 Exercises

3.1 Let X denote a real-valued random variable with a discrete distribution with frequency function
x
p(x) = θ(1 − θ) , x = 0, 1,... ;
where θ is a fixed constant, 0 <θ < 1. Find the characteristic function of X.