Theory of Statistics - George Mason University

1.1 Some Important Probability Facts 49
Proof. (Theorem 1.17)
We are given the finite scalar moments ν0, ν1, . . . (about any origin) of some
probability distribution, and the condition that
∞
j=0
νjt j
j!
converges for some real nonzero t. We want to show that the moments uniquely
determine the distribution. We will do this by showing that the moments
uniquely determine the CF.
Because the moments exist, the characteristic function ϕ(t) is continuous
and its derivatives exist at t = 0. We have for t in a neighborhood of 0,
ϕ(t) =
r
j=0
(it) j µj
j! + Rr, (1.100)
where |Rr| < νr+1|t| r+1 /(r + 1)!.
Now because νjtj /j! converges, νjtj /j! goes to 0 and hence the right
hand side of equation (1.100) is the infinite series ∞ (it)
j=0
j µj
if it con-
j!
verges. This series does indeed converge because it is dominated termwise
by νjtj /j! which converges. Thus, ϕ(t) is uniquely determined within a
neighborhood of t = 0. This is not sufficient, however, to say that ϕ(t) is
uniquely determined a.e.
We must extend the region of convergence to IR. We do this by analytic
continuation. Let t0 be arbitrary, and consider a neighborhood of t = t0.
Analogous to equation (1.100), we have
ϕ(t) =
r
j=0
ij (t − t0) j ∞
j!
−∞
x j e it0x dF + Rr.
Now, the modulus of the coefficient of (t − t0) j /j! is less than or equal to νj;
hence, in a neighborhood of t0, ϕ(t) can be represented as a convergent Taylor
series. But t0 was arbitrary, and so ϕ(t) can be extended through any finite
interval by taking it as the convergent series. Hence, ϕ(t) is uniquely defined
over IR in terms of the moments; and therefore the distribution function is
uniquely determined.
Characteristic Functions for Functions of Random Variables;
Joint and Marginal Distributions
If X ∈ IR d is a random variable and for a Borel function g, g(X) ∈ IR m , the
characteristic function of g(X) is
ϕg(X)(t) = EX
Theory of Statistics c○2000–2013 James E. Gentle
e itT g(X)
, t ∈ IR m , (1.101)