Theory of Statistics - George Mason University

750 0 Statistical Mathematics
A linear transform with ψ(s, x) ∝ (e sx ) c for some c, such as the Fourier
transform, the Laplace transform, and the characteristic function, satisfies the
“change of scale property”:
T (f(ax))(s) = 1
s
T (f(x)) , (0.1.106)
|a| a
where a is a constant. This is easily shown by making a change of variables
in the definition (0.1.104). This change of variables is sometimes referred to
as “time scaling”, because the argument of f often corresponds to a measure
of time. A similar scaling applies to the argument of the transform f T , which
is sometimes called “frequency scaling”.
Transforms in which ψ(s, x) ∝ (e sx ) c also have two useful translation properties:
• for a shift in the argument of f,
T (f(x − x0))(s) = ψ(s, x0)T (f(x))(s) : (0.1.107)
• for a shift in the argument of the transform T f,
T (f(x))(s − s0) = T (ψ(−s0, x)f(x))(s). (0.1.108)
These scaling and translation properties are major reasons for the usefulness
of the Fourier and Laplace transforms and of the characteristic function in
probability theory.
Linear transforms apply to multivariate functions as well as to univariate
functions. In the definition of linear transforms (0.1.104), both s and x
may be vectors. In most cases s and x are vectors of the same order, and
specific transforms have simple extensions. In the characteristic function of
multivariate random variable, for example,
Fourier Transforms
ψ(s, x) = e i〈s,x〉 .
The Fourier transform of a function f(x) is the function
Ff(s) =
∞
if the integral exists.
The inverse Fourier transform is
f(x) =
∞
−∞
−∞
e 2πisx f(x)dx, (0.1.109)
e −2πisx Ff(s)ds. (0.1.110)
Instead of e 2πisx as in equation (0.1.109), the Fourier transform is often
defined with the function e iωx , in which ω is called the “angular frequency”.
Theory of Statistics c○2000–2013 James E. Gentle