On the free L´evy measure of the normal distribution

Belinschi et al. [5] proved that the normal distribution is freely inﬁnitely divisible. This paper establishes a certain monotonicity, real analyticity and asymptotic behavior of the density of the free L´evy measure. The monotonicity property strengthens the result in Hasebe et al. [13] that the normal distribution is freely selfdecomposable.


Introduction 1.Backgrounds
The role of the normal distribution is played by Wigner's semicircle distribution in free probability.Most notably, the latter appears in the free central limit theorem (see e.g.[14,17,22]).Although a role of the normal distribution in free probability is not very obvious, there are still some attempts to understand it.In [5] the normal distribution was proven to be freely infinitely divisible, and then, as a stronger result, the normal distribution was proven to be freely selfdecomposable in [13].Combinatorial aspects are also investigated in [5].
This paper further analyzes the free infinite divisibility of the normal distribution.Key analytical machineries are the Cauchy transform, its reciprocals and the Voiculescu transform, defined as follows.The Cauchy transform of a probability measure µ on R is the function It is easy to see that G µ is analytic and maps the complex upper half-plane (denoted C + ) to the lower half-plane (denoted C − ) and also C − to C + .Note that the Cauchy transform is often defined only on C + but in this paper the values on C − is also useful, see (2.2).Then the reciprocal Cauchy transform of µ is defined to be which is an analytic selfmap of C + .
A probability measure µ on R is said to be freely infinitely divisible if for any n ∈ N there exists a probability measure µ n on R such that A basic fact is a characterization of freely infinitely divisible distributions in terms of the Voiculescu transform.
Theorem 1.1 ([7, Theorem 5.10]).A probability measure µ on R is freely infinitely divisible if and only if the Voiculescu transform ϕ µ has an analytic extension defined on C + with values in C − ∪ R.
For a freely infinitely divisible distribution µ, let ϕ µ denote the analytic extension of its Voiculescu transform as described above.Then the transform ϕ µ has the following Pick-Nevanlinna representation: for some b µ ∈ R and a finite measure τ µ on R. The pair (b µ , τ µ ) is unique.The measure is called the free Lévy measure of µ and the mass of τ µ at zero is called the semicircular component.The standard semicircle distribution (i.e. with mean 0 and variance 1) corresponds to (b µ , τ µ ) = (0, δ 0 ).For many classical distributions including the normal distribution, its Voiculescu transform cannot be explicitly calculated.In such a case, the following condition has been a useful sufficient (but not necessary, see [3,Proposition 3.6] for a counterexample) condition for proving the free infinite divisibility.
Many classical distributions, despite their unexplicit Voiculescu transforms, have been proven to be in class UI.They include the normal distribution [5], some beta distributions and some gamma distributions [10] and some HCM distributions [11]; see e.g.[5,4,8,10,11,16] for further examples.On the other hand, little is known about free Lévy measures of these distributions; most of the former results in the literature were limited to the existence of Ω.For the normal distribution, it is nonetheless shown in [13] that the free Lévy measure of N(0, 1) is of the form where k is nondecreasing on (0, ∞) and is non-increasing on (−∞, 0), i.e., N(0, 1) is freely selfdecomposable.The fact that N(0, 1) is symmetric also implies that k is an even function.The aim of this paper is to clarify further properties of the function k.

Main results and the outline of proofs
The main result of this paper is the following two theorems on the free Lévy measure of N(0, 1).
The function h will be defined in (2.11); it describes the height of the boundary of the domain Ω introduced in Definition 1.2 for the normal distribution N(0, 1).Therefore, studying the free Lévy measure is closely related to studying the boundary of Ω.
We then clarify the asymptotic behavior of the function h of the normal distribution at zero and at infinity.
Remark 1.6.Moreover, (h ∞ ) can be enhanced to the asymptotic expansion see Remark 3.4 for further details.
Proofs of the above two theorems are sketched here.First we construct an entire analytic continuation G of the Cauchy transform of N(0, 1) (Subsection 2.1).Then we introduce a crucial supplementary domain Ξ ⊆ C on which the reciprocal Cauchy transform F := 1/ G is analytic (Lemma 2.4; note that F is a meromorphic function on C).Moreover, a simply connected domain Ω for the normal distribution as in Definition 1.2 exists as a subset of Ξ (Theorem 2.8).In the construction of Ω, we first identify its boundary set ∂Ω with the preimage of R \ {0} by the map F ↾ Ξ (Proposition 2.6).(If we go outside of Ξ, then the preimage of R \ {0} seems to have irrelevant connected components, see Figure 3 below.)This argument also implies the analyticity of ∂Ω.
A key fact is that ∂Ω is a graph of a function.The height (from the real line) of the boundary curve ∂Ω can be described by the function h(x) := −Im[( F ↾ cl(Ω) ) −1 (x)], x > 0, which is exactly the function appearing in Theorem 1.4.According to [15, (8.1.6)]or [5, (3.5)], the following ODE is satisfied by F N (0,1) : By analytic continuation, this formula holds for F on Ξ too.Going to the inverse map, a system of ODEs for h(x) (and for g(x) ) can be deduced.The monotonicity of h in Theorem 1.4 is an easy consequence of these ODEs (Proposition 2.7).Formula (1.4) easily follows from the Stieltjes inversion (Section 3).Considering the above, investigating the curve ∂Ω in further details will reveal fine properties of the free Lévy measure, which results in Theorem 1.5.The two asymptotics (h ∞ ) and (h 0 ) will be separately proved in expanded forms (providing finer descriptions of ∂Ω) as Theorems 3.3 and 3.5, respectively.The proof does not use the ODE by contrast to Theorem 1.4.
The method for proving (h ∞ ) is based on asymptotic analyses of the reciprocal Cauchy transform and of its inverse function at infinity on a region | arg z| < ε.As a basis, the Laurent series asymptotic expansions of F and its inverse are obtained in Lemmas 2.2 and 3.2, respectively.In addition, we need to estimate an exponential decay of Im[ F ], which is invisible in the Laurent series expansion (Proof of Theorem 3.3).This decay is inherited from the tail behavior of the probability density function.The whole method seems to be applicable to a wider class of freely infinitely distributions with unbounded support.
The proof of (h 0 ) is based on formula (2.2) for G.As the curve ∂Ω goes to infinity as it approaches the negative imaginary axis (cf. Figure 3), the contribution of G(z) in formula (2.2) is negligible because of its order O(1/z) on ∂Ω (and because the remaining term A key observation in this analysis is that ∂Ω can be well approximated by the curve ∂Ξ near the imaginary axis, cf. Figure 3.

The Cauchy transform of the normal distribution
In this section, we analyze the analytic continuation of the Cauchy transform and its reciprocal, and then describe the boundary of Ω as a graph of an analytic function, where Ω is the domain appearing in Definition 1.2 for N(0, 1).

Entire analytic continuation of the Cauchy transform
We simplify the notation of the Cauchy transform of the normal distribution into A well known fact is that the Cauchy transform G↾ C + has an analytic continuation to C (denoted by G) and, on the lower half-plane, the formula holds, see e.g.[9, Theorem 1.2].On the other hands, due to [9, p. 362] and the identity theorem, we have In particular, we have which can also be deduced from the Stieltjes inversion formula.
Obviously, the reciprocal Cauchy transform F N (0,1) = 1/G N (0,1) analytically extends to the meromorphic function on C In view of (1.1) and (2.4), poles of F do not exist in C + ∪ R. It seems that F has poles on C − , see Figure 5 below; however, we mostly work on F in subdomains where F turns out to have no poles, so that analysis of poles will be rather out of scope of this paper.

Behavior of the Cauchy transform on extended domains
As preparatory steps, we investigate the behavior of G and F on the imaginary axis (Lemma 2.1), asymptotic behavior as (2) F is a bijection from iR onto i(0, ∞).
Proof.Obviously it suffices to verify the equivalent assertions Proof of (i) and (ii).
The Cauchy transform is well known to have an asymptotic expansion as z → ∞, ε < arg z < π−ε for any fixed ε ∈ (0, π), see e.g.[1, Theorem 3.2.1].For the normal distribution, we can see that the asymptotic expansion holds in the larger domain with ε ∈ (0, π/4) arbitrary but fixed.To state the formula, we denote by {m k } k≥0 the moment sequence of N(0, 1), i.e.
For any fixed N ∈ N and fixed ε ∈ (0, π 4 ), the asymptotic expansions Proof.For 0 < ε < π/4 we first observe that This is an easy consequence of Cauchy's integral formula applied to the region(s) in Figure 1 and the fact that the contour integrals over the arcs The integral over A 2 R is similarly estimated.The remaining arguments are analogous to the standard one for ε < arg z < π − ε, see e.g. the proof of [1, Theorem 3.2.1].For the reader's convenience the rest of the proof is included in Appendix A.
By Lemma 2.2, the analytic extension F has no poles on D ε,R := D ε ∩{z : |z| > R} for sufficiently large R > 0 and (2.9) The next two lemmas are basic ingredients to construct and analyze a compositional inverse function of F .
which implies that the curve F (∂D ε,R ) does not intersect with D ε ′ ,R ′ and every point of D ε ′ ,R ′ has rotation number 1 with respect to this curve (viewed as a closed curve in the Riemann sphere), and hence Part 2: Univalence of F .In order to resort to the Noshiro-Warschawski criterion (see e.g.[20,Proposition 1.10] or the original articles [18,23]), we estimate the derivative F ′ on D ε,R .Take 0 < η < ε < π/4.By Lemma 2.2, we have and therefore, we can take R 0 > 0 large enough so that Re[ Because D ε,R 0 is not convex, we introduce supplementary convex domains.Let ℓ be the half-line starting from the point 4R 0 e i(− π 4 +ε) , passing 4R 0 i and going to infinity.Let U be the domain that has the boundary ℓ ∪ {re i(− π 4 +ε) : r ≥ 4R 0 } and contains the point 4R 0 (1 + i).Let V be the reflection of U with respect to the imaginary axis.Since U and V are convex domains contained in D ε,R 0 , the Noshiro-Warschawski criterion implies that F is univalent in U and V .Choosing R = 8R 0 and using the fact that F is close to the identity map, i.e.
Proof of (i)-(iii).The proofs are based on separate analyses of the two terms of formula (2.2).

Construction of Ω and description of its boundary
Using results in the previous subsection, we construct Ω and describe its boundary (Theorem 2.8).The boundary turns out to be the graph of an analytic function.We first provide supplementary facts.
Proof.Let c R be the simple closed curve in the Riemann sphere consisting of i[−∞, 0], {z ∈ ∂Ξ : 0 < Re(z) < R}, {R + iy : − π 2R ≤ y ≤ 0} and [0, R].We take R sufficiently large so that | F (z) − z| < 1 2 |z| for |z| ≥ R, z ∈ c R .By Lemma 2.1, Lemma 2.4 (1), (3) and the fact that Im[F ] > 0 on R, we can observe that every point of (0, 1  2 R) is surrounded by the curve F (c R ) exactly once.This implies that for every x ∈ (0, 1  2 R) there exists a unique point H(x) in the Jordan domain surrounded by c R such that F (H(x)) = x.Because R is arbitrary as long as sufficiently large, for every x ∈ (0, ∞) there exists a unique point H(x) in the domain surrounded by the curve i[−∞, 0] ∪ {z ∈ ∂Ξ : 0 < Re(z) < ∞} ∪ [0, ∞) such that F (H(x)) = x.It remains to prove the analyticity of the function H. First note that F ′ (z) = 0 holds on p + 0 ; otherwise the point H(x) would not be unique.Therefore, F is locally bijective at each point H(x) and hence its inverse function H is also analytic in a complex neighborhood of each point x > 0.
We then study properties of analytic functions Note that this function h will turn out to coincide with h in Theorem 1.4, see (3.1) and (3.3).Obviously, we have g, h > 0. By analytic continuation, Equation (1.5) easily extends to Because H is the compositional inverse map of F ↾ p + 0 and F ′ (z) does not vanish on p + 0 , the ODE (2.12) restricted to p + 0 entails the ODE H ′ (x) = 1 x(H(x)−x) , which is equivalent to and (2.13) Using this ODE we provide some properties of g and h below.Some of the results will be made much finer in Section 3.
It remains to prove the last two limits.It suffices to establish lim x→0 + h(x) = ∞, because then lim x→0 + g(x) = 0 follows from the fact that g(x) − ih(x) ∈ Ξ.
Proof.First note that Ω is exactly the domain that has boundary p + 0 ∪ p − 0 and contains C + as a subset.
Because ∂Ξ ∪ {∞} is not a simple closed curve in the Riemann sphere, we introduce an approximating simple closed curve γ R in the Riemann sphere consisting of {z ∈ p + 0 ∪{∞}∪p − 0 : |Re(z)| ≤ R}, the semicircle {Re iθ : 0 ≤ θ ≤ π} and two vertical line segments R+i[−f (R), 0] and −R+i[−f (R), 0], where ∞ stands for the point corresponding to lim x→0 + H(x) and R > 0. Note that, as a consequence of Lemma 2.4 (3), F can be regarded as a continuous function on closure (in the Riemann sphere) of the Jordan domain surrounded by γ R .This fact is important in the next paragraph when we apply the Darboux theorem.
We prove that F is univalent on γ R for sufficiently large R > 0. Because of the symmetry with respect to the imaginary axis, it suffices to prove the univalence on γ R ∪ {z : Re(z) > 0}.First, F is univalent on p + 0 by the construction of p + 0 .Also, we can take r, R > 0 with r, g(r) < R/4 large enough so that, by Lemma 2.3, F is univalent in γ R ∩ {x + iy : |x| ≥ g(r) or y ≥ r} and that, by (2.9), | F (z) − z| < |z| 2 holds for all |z| ≥ R, z ∈ γ R .In this situation one can see the univalence on the whole γ R .This furthermore implies that F is a bijection from the Jordan domain surrounded by γ R onto the Jordan domain surrounded by F (γ R ) according to the Darboux theorem, see e.g.[20, or [19,Corollary 9.5]. 1 By letting R → ∞, we conclude that F is an analytic bijection from Ω onto C + .
Finally, by Carathéodory's theorem [20,Theorem 2.6] and the fact that γ R is a Jordan curve in the Riemann sphere, F is a homeomorphism from cl(Ω) onto (C + ∪ R) \ {0}.Remark 2.9.Theorem 2.8 implies the fact N(0, 1) ∈ UI that follows easily from the work [5], where probability measures µ c , c ∈ (−1, 0), called the Askey-Wimp-Kerov distributions, are shown to be in UI.Because class UI is weakly closed (see [2, p. 2763]) and N(0, 1) is the weak limit of µ c as c → 0 − , we conclude that N(0, 1) also belongs to UI.The strategy there for proving µ c ∈ UI was to construct, for each t > 0, a simple curve p c t which is symmetric with respect to the imaginary axis, passing through a unique point in iR and so that the reciprocal Cauchy transform maps p c t bijectively onto R + it, see [5,Lemma 3.8] for the construction of p c t .However, the domain Ω was not investigated in [5].By contrast, our method directly constructed the boundary p + 0 ∪ p − 0 of Ω as the preimage of R \ {0} by the map F ↾ Ξ (for the interested reader, the curves Im[ F ] = t for different t's are shown in Figure 4).Further details on the boundary p + 0 ∪ p − 0 will be clarified in Theorems 3.3 and 3.5 below.

Proofs of the main results
According to Theorem 2.8, we can define the analytic compositional inverse function according to the previous section and by symmetry.On the other hand, by Lemma 2.3, the function F is a bijection from D ε,R onto its range that contains D ε ′ ,R ′ for sufficiently large R.This allows us to get an analytic continuation of F −1 to the domain ( Proof of Theorem 1.4.Recall that the Voiculescu transform ϕ(w) where b ∈ R and τ is a finite measure on R, the Stieltjes inversion formula (see e.g.[21, Theorems F.2, F.6]) yields According to (1.2) the free Lévy measure of N(0, 1) is given as desired.Let κ n be the n-th free cumulant of N(0, 1) below.Because N(0, 1) is symmetric, the odd free cumulants vanish.According to [5, p. 3683], some even free cumulants are given as follows: The following fact is a key for understanding the asymptotics of h(x) as x → ∞.We can prove it analogously to [6,Theorem 1.3 and Proposition A.3].For the reader's convenience a self-contained proof is provided in Appendix A. Lemma 3.2.For every fixed N ∈ N and ε ′ ∈ (0, π/4) we have We provide a proof of Theorem 1.5 (h ∞ ) below.Since the proof requires estimates on g too, we expand the statement of Theorem 1.5 (h ∞ ) as follows.
Proof.(g ∞ ) is just the real part of the formula in Lemma 3.2.
For the proof of (h ∞ ), we set a(x, y) := Im[ F (x + iy)].It follows from (2.4) and Lemma 2.2 that, as x → ∞, as long as x + iy ∈ D ε .By Taylor's theorem, for every x > 0 and y ∈ [−1, 0] there exists θ ∈ (0, 1) such that a(x, y) = a(x, 0) + a y (x, θy)y, so that If we set the function where the O(x −2 )'s are all independent of c.This implies, for a sufficiently large (fixed) c > 0, that for all sufficiently large x (such that f c (x) > −1).From the results in Subsection 2.3, for x > 0, the function f (x) introduced in Theorem 2.8 is a unique solution y ∈ (−π/(2x), 0) to the equation a(x, y) = 0. We therefore have Finally, using g( Remark 3.4.With some more elaboration, (h ∞ ) can be generalized to higher order expansions of any order where the coefficients a 2n are determined by the formula (in the sense of formal power series or asymptotic expansion) The proof is sketched below.We first refine (3.5) and (3.6).From Lemma 2.2 we have where for some constants b 2n ∈ R (n ∈ N) (the Boolean cumulants of N(0, 1)).Then we get the following refinement of (3.5): On the other hand, for (3.6), we restrict the variables (x, y) to the thin domain The point is that x + if (x) is contained in J for large x thanks to the established (3.7).Then we can get the following refinement of (3.6): x → ∞ with x + iy ∈ J. (3.12) The point is that no y's appear in the main term above thanks to the exponential bounds for y.
Then substituting y = f (x) into a(x, y) = a(x, 0) + a y (x, θy)y and combining it with (3.11) and (3., the right hand side of which can be written as F (x) 2 x 2 F ′ (x) .Finally computing h(x) = −f (g(x)) as in (3.8) yields the desired formula (3.9).Note that we can write g(x) = F −1 (x) in the sense of asymptotic expansion, which is a key ingredient for deriving (3.10).
Next, we provide a proof of Theorem 1.5 (h 0 ).The proof needs estimates on the functions f , g and gh, so we expand the statement of Theorem 1.5 (h 0 ).The results also offer a better understanding of the boundary of Ω; especially they justify that the curve {H(x) : x > 0} approaches ∂Ξ as x → 0 + as observed in Figure 3.In particular, h(x) ∼ 2 log 1 x .