APPENDIX F
Characteristic Functions
ddddd, dddddddddddddddddddddd. dddddddd
dddd probability measure. dd probability measure ddddddddd. ddd
dddd probability measure dd Fourier transform ddddddd.
F.1. General properties
Recall: X is a random variable with distribution function F . Then
μX (x) = P(X ≤ x) = F (x).
Moreover,
E[X] =
Ω
X dP =
y μX (dy) =
R1
∞
−∞
y dF (y).
(1) Suppose that μ is probability measure on (R1 , B 1 ). Then
Definition F.1.
the characteristic function (ch.f.) of μ is defined by
μ̂(t) =
R1
eitx μ(dx)
=
R1
cos(tx) μ(dx) + i
R1
sin(tx) μ(dx).
(2) Let F be the distribution function on R1 . Then the characteristic function of F
is defined by
F̂ (t) =
∞
−∞
eitx dF (x).
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336
F. CHARACTERISTIC FUNCTIONS
(3) Let X be a random variable. Then the characteristic function of X is defined by
μ̂X (t) =
R1
itx
e
μX (dx) =
∞
−∞
eitx dFX (x) = E[eitX ]
dddddddddd, ddddddd probability measure, random variable d
distribution function ddddddd, ddddddd.
Proposition F.2.
(1) μ̂ : R1 −→ C is a continuous, complex-valued function.
(2) μ̂(0) = 1
|μ̂(t)| ≤ 1
μ̂(−t) = μ̂(t)
(3) If (fn )n≥1 is a sequence of characteristic functions, λn ≥ 0,
∞
∞
λn = 1, then
n=1
λn fn is a characteristic function.
n=1
(4) If f1 and f2 are characteristic functions, then f1 · f2 is a characteristic function.
(This implies that the product of finite characteristic functions is a characteristic
function.)
(5) If f is a characteristic function, then f¯ is a characteristic function.
(If f is the characteristic function of X, then f¯ is a characteristic function −X.)
(6) If f is a characteristic function, then |f |2 is a characteristic function.
(1) Suppose that X is normally distributed N (m, σ 2 ), then its
σ 2 t2
characteristic function is given by exp(imt −
).
2
a
(2) X is Cauchy distributed, i.e., the density function of X =
, then
2
π(a + (x − θ)2 )
its characteristic function is given by exp(iθt − a|t|)
Example F.3.
F.2. UNIQUENESS AND INVERSION
337
(3) Suppose that X is uniform distributed on [−a, a]. Then its characteristic function
is given by
⎧
⎪
sin(at)
⎪
⎨
,
at
⎪
⎪
⎩1,
if t = 0,
if t = 0.
(4) X is Poisson distributed with distribution function, i.e, F has the distribution
function
F (x) =
[x]
e−λ
k=0
λk
.
k!
Then its characteristic function = exp[λ(eit − 1)].
dd Fourier transform dd, ddddddddddd, dddddddddddd
dd.
F.2. Uniqueness and inversion
dddddddddd characteristic function d 1 − 1 dd, dddddddddd
ddddddd measure ddd, dddd functions ddd, dddddd 1 − 1 dd
ddd, ddddd function ddddd measure d.
Theorem F.4. Let μ be a probability measure and let f be its characteristic function
(i.e., f (t) = μ̂(t)). Then for x1 < x2 , we have
1
1
1
μ((x1 , x2 )) + μ({x1 }) + μ({x2 }) = lim
T −→∞ 2π
2
2
T
−T
e−itx1 − e−itx2
f (t) dt.
it
(ddddd inversion formula)
Theorem F.5.
(1) If μ({x1 }) = μ({x2 }) = 0
1
μ((x1 , x2 )) = lim
T −→∞ 2π
T
−T
e−itx1 − e−itx2
f (t) dt.
it
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F. CHARACTERISTIC FUNCTIONS
(2) For x0 ∈ R1 .
1
μ({x0 }) = lim
T −→∞ 2π
T
−T
e−itx0 f (t) dt.
Corollary F.6. If μ1 , μ2 ∈ M1 , and μ̂1 (t) = μ̂2 (t) for all t, then μ1 ≡ μ2 , i.e., the
characteristic function determines probability measure.