Letχ(u)denote the characteristic function of an infinitely divisible distribution. Thenχ(u)possesses a Lévy-Khintchine representation.
χ(u) = exp
iubưu2 2 c+
Z
IR\{0} eiux−1−iuh(x) K(dx)
. (2.1)
(See also Chapter 1.) Hereb ∈IRand c ≥ 0are constants, andK(dx) is the Lévy measure. This is a σ-finite measure onIR\{0}that satisfies
Z
IR\{0}(x2∧1) K(dx)<∞. (2.2)
It is convenient to extendK(dx)to a measure onIRby settingK({0}) = 0. Unless stated otherwise, by K(dx)we mean this extension. The functionh(x)is a truncation function, that is, a measurable bounded function with bounded support that that satisfiesh(x) =xin a neighborhood ofx= 0. (See Jacod and Shiryaev (1987), Definition II.2.3.) We will usually use the truncation function
h(x) =x1l{|x|≤1}.
3The author has developed an S-Plus program based on this method. This was used by Wiesendorfer Zahn (1999) for the simulation of hyperbolic Lévy motions.
The proofs can be repeated with any other truncation function, but they are simpler with this particular choice ofh(x).
In general, the Lévy measure may have infinite mass. In this case the mass is concentrated aroundx= 0.
However, condition (2.2) imposes restrictions on the growth of the Lévy measure aroundx= 0.
Definition 2.1. Let K(dx) be the Lévy measure of an infinitely divisible distribution. Then we call modified Lévy measure the measureKe on(IR,B)defined byK(dx) :=e x2K(dx).
Lemma 2.2. Let Ke be the modified Lévy measure corresponding to the Lévy measure K(dx) of an infinitely divisible distribution that possesses a second moment. ThenKe is a finite measure.
Proof. Since x 7→ x2 ∧1 is K(dx) integrable, it is clear that Ke puts finite mass on every bounded interval. Moreover, by Wolfe (1971), Theorem 2, if the corresponding infinitely divisible distribution has a finite second moment,x2 is integrable over any whose closure does not containx = 0. SoKe assigns finite mass to any such set. Hence
Ke(IR) =Ke [−1,1]
+Ke (−∞,−1)∪(1,∞)
<∞.
The following theorem shows how the Fourier transform of the modified Lévy measure x2K(dx) is connected with the characteristic function of the corresponding distribution. This theorem is related to Bar-Lev, Bshouty, and Letac (1992), Theorem 2.2a, where the corresponding statement for the bilateral Laplace transform is given.4
Theorem 2.3. Let χ(u) denote the characteristic function of an infinitely divisible distribution onIR possessing a second moment. Then the Fourier transform of the modified Lévy measure x2K(dx) is given by
Z
IR
eiuxx2K(dx) =−c− d du
χ0(u) χ(u)
. (2.3)
Proof. Using the Lévy-Khintchine representation, we have d
duχ(u) =χ(u)ã
ibưuc+ d du
Z
IR
eiux−1−iuh(x) K(dx)
. The integrandeiux−1−iuh(x)is differentiable with respect tou. Its derivative is
∂u eiux−1−iuh(x)
=ix eiux−ih(x).
This is bounded by aK(dx)-integrable function as we will presently see. First, for|x| ≤1we have
|ix eiux−ih(x)|=|x| ã |eiux−1|
≤ |x| ã(|cos(ux)−1|+|sin(ux)|)
≤ |x| ã2|ux|=|u| ã |x|2.
4However, Bar-Lev, Bshouty, and Letac (1992) do not give a proof. They say “The following result does not appear clearly in the literature and seems rather to belong to folklore.”
Forufrom some bounded interval, this is uniformly bounded by some multiple of|x|2. For|x|>1,
|ix eiux−ih(x)|=|ix eiux|=|x|. From Wolfe (1971), Theorem 2, it follows thatR
{|x|>1}|x|K(dx) < ∞ iff the distribution possesses a finite first moment. Hence for eachu∈IRwe can find some neighborhoodUsuch thatsupu∈U|ix eiux− ih(x)|is integrable, Therefore the integral is a differentiable function ofu, and we can differentiate under the integral sign. (This follows from the differentiation lemma; see e. g. Bauer (1992), Lemma 16.2.) Consequently, we have
χ0(u)
χ(u) =ibưuc+ Z
IR
ix eiux−ih(x)
K(dx).
Again by the differentiation lemma, differentiating a second time is possible if the integrand ix eiux− ih(x)has a derivative with respect touthat is bounded by someK(dx)-integrable functionf(x), uni- formly in a neighborhood of anyu∈IR. Here this is satisfied withf(x) =x2, since
∂
∂u ix eiux−ih(x)=| −x2 eiux|=x2 for allu∈IR.
Again by Wolfe (1971), Theorem 2, this is integrable with respect toK(dx) because by assumption the second moment of the distribution exists. Hence we can again differentiate under the integral sign, getting
d du
χ0(u) χ(u)
=−c+ Z
IR
eiuxãx2 K(dx).
This completes the proof.
Corollary 2.4. Letχ(u)be the characteristic function of an infinitely divisible distribution on(IR,B) that integratesx2. Assume that there is a constantec∈IRsuch that the function
b
ρ(u) :=−ec− d du
χ0(u) χ(u) (2.4)
is integrable with respect to Lebesgue measure. Thenecis equal to the Gaussian coefficientcin the Lévy- Khintchine representation, and ρ(u)b is the Fourier transform of the modified Lévy measurex2K(dx).
This measure has a continuous Lebesgue density onIRthat can be recovered from the functionρ(u)b by Fourier inversion.
x2dK
dλ(x) = 1 2π
Z
IR
e−iuxρ(u)du.b
Consequently, the measureK(dx)has a continuous Lebesgue density onIR\{0}:
dK
dλ(x) = 1 2πx2
Z
IR
e−iuxρ(u)du.b For the proof, we need the following lemma.
Lemma 2.5. LetG(dx)be a finite Borel measure onIR. Assume that the characteristic functionG(u)b ofGtends to a constantcas|u| → ∞. ThenG({0}) =c.
Proof of Lemma 2.5. For any Lebesgue integrable function g(u) with Fourier transform bg(x) = R eiuxg(u)du, we have by Fubini' s theorem that
Z b
g(x)G(dx) = Z Z
eiuxg(u)du G(dx)
= Z Z
eiuxG(dx)g(u)du=
Z G(u)g(u)b du.
(2.5)
Setting ϕ(u) := (2π)−1/2e−u2/2, we get the Fourier transformϕ(x) =b e−x2/2. Now we consider the sequence of functionsgn(u) :=ϕ(u/n)/n, n≥1. We havegbn(x) =ϕ(nx)b → 1l{0}(x)asn→ ∞, for anyx∈IR. By dominated convergence, this implies
Z b
gn(x)G(dx)→ Z
1l{0}(x)G(dx) =G({0}) (n→ ∞).
(2.6)
On the other hand, settingGbn(u) :=G(nu),b n≥1, we haveGbn(u) →1l{x=0}+c1l{x6=0} pointwise foru∈IR. Hence, again by dominated convergence,
Z G(u)gb n(u)du=
Z G(u)ϕ(u/n)/n dub
=
Z G(nu)ϕ(u)b du→ Z
(1l{x=0}+c1l{x6=0})ϕ(u)du=c.
(2.7)
Since we have R b
gn(x) G(dx) = R bG(u)gn(u) du by (2.5), now relations (2.6) and (2.7) yield the desired result:
R gbn(x)G(dx) (2.6)−→ G {0} R bG(u)g||n(u)du (2.7)−→ c.
Proof of Corollary 2.4. By Lemma 2.2,x2K(dx)is a finite measure under the hypotheses of Corollary 2.4. By Theorem 2.3, its Fourier transform is given by
Z
IR
eiuxx2K(dx) =−c− d du
χ0(u) χ(u)
.
On the other hand, the assumed integrability ofρ(u)b implies thatρ(u)b →0as|u| → ∞. Soρ(u) +b ec−c, which is just the Fourier transform ofx2K(dx), converges to the valueec−cas|u| → ∞. Lemma 2.5 now yields that the limitec−cis just the modified Lévy measure of the set{0}. But this is zero, so indeed e
c=c.
The remaining statements follow immediately from Theorem 2.3 and the fact that integrability of the Fourier transform implies continuity of the original function. (See e. g. Chandrasekharan (1989), I.(1.6).)