Page 89 - Elements of Distribution Theory

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

3.2 Basic Properties 75

Then h is of bounded variation and for any a < b
b b b

|h(y)| dy = F(x + y) dy − F(y) dy
a a a
b+x a+x
= F(z) dz − F(z) dz
b a
≤ [F(b + x) − F(a)]x ≤ x.
Hence,

∞
|h(y)| dy < ∞.
−∞
It follows from the Theorem 3.2 that
1 T ∞
h(y) = lim exp{−ity} h(z)exp(itz) dz dy
2π T →∞ −T −∞
provided that h is continuous at y; note that, in this expression, the integral with respect to
dz is simply the Fourier transform of h. Consider the integral
∞ ∞

1
h(z)exp{itz} dz = h(z) d exp{itz}.
it
−∞ −∞
Using integration-by-parts, this integral is equal to
1 ∞ exp{itz} ∞

− exp{itz} dh(z) + h(z) .
it −∞ it −∞
Note that exp{itz} is bounded,
lim h(z) = lim h(z) = 0,
z→∞ z→−∞
and

∞ ∞ ∞
exp{itz} dh(z) = exp{itz} dF(x + z) − exp{itz} dF(z)
−∞ −∞ −∞
= ϕ(t)[exp{−itx}− 1].
Hence,
∞ 1 − exp(−itx)

h(z)exp(itz) dz = ϕ(t) ,
it
−∞
so that
1 T exp{−ity}− exp{−it(x + y)}
h(y) = lim ϕ(t) dt,
2π T →∞ −T it
provided that h is continuous at y, which holds provided that F is continuous at x and
x + y. Choosing y = x 0 and x = x 1 − x 0 yields the result.

Thus, given the characteristic function of a random variable X we may determine differ-
ences of F, the distribution function of X,of the form F(x 1 ) − F(x 0 ) for continuity points
x 0 , x 1 .However, since set of points at which F X is discontinuous is countable, and F is
right-continuous, the characteristic function determines the entire distribution of X. The
details are given in the following corollary to Theorem 3.3.