Page 102 - Elements of Distribution Theory

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88 Characteristic Functions

Hence, the distribution of Y is symmetric about 0; the distribution of Y is called the sym-
metrization of the distribution of X.
For instance, suppose that X has a uniform distribution on the interval (0, 1). Then Y
has characteristic function
2
sin(t) + (cos(t) − 1) 2
ϕ Y (t) = , −∞ < t < ∞;
t 2
see Example 3.14.

Lattice distributions
Let X denote a real-valued discrete random variable with range
X ={x 1 , x 2 ,...}

where x 1 < x 2 < ··· and Pr(X = x j ) > 0, j = 1, 2,.... The distribution of X is said to
be a lattice distribution if there exists a constant b such that, for any j and k, x j − x k is a
multiple of b. This occurs if and only if X is a subset of the set
{a + bj, j = 0, ±1, ±2,...}

for some constant a; b is said to be a span of the distribution. A span b is said to be a
maximal span if b ≥ b 1 for any span b 1 .
Stated another way, X has a lattice distribution if and only if there is a linear function of
X that is integer-valued.

Example 3.17 (Binomial distribution). Consider a binomial distribution with parameters
n and θ. Recall that range of this distribution is {0, 1,..., n} with frequency function

n x n−x
p(x) = θ (1 − θ) , x = 0, 1,..., n.
x
Hence, this is a lattice distribution with maximal span 1.

Example 3.18 (A discrete distribution that is nonlattice). Let X denote a real-valued
√
random variable such that the range of X is X ={0, 1, 2}. Suppose this is a lattice
distribution. Then there exist integers m and n and a constant b such that
√
2 = mb and 1 = nb.
√ √
It follows that 2 = m/n for some integers m, n. Since 2is irrational, this is impossible.
It follows that the distribution of X is non-lattice.
More generally, if X has range X ={0, 1, c} for some c > 0, then X has a lattice
distribution if and only if c is rational.

The characteristic function of a lattice distribution has some special properties.

Theorem 3.11.
(i) The distribution of X is a lattice distribution if and only if its characteristic function
ϕ satisfies |ϕ(t)|= 1 for some t = 0. Furthermore, if X has a lattice distribution,
then |ϕ(t)|= 1 if and only if 2π/tisa span of the distribution.