Unimodality of Boolean and monotone stable distributions

We give a complete list of the Lebesgue-Jordan decomposition of Boolean and monotone stable distributions and a complete list of the mode of them. They are not always unimodal.


Unimodality of Boolean and monotone stable distributions
First, we gather analytic tools and their properties to compute Boolean and monotone stable distributions.

Analytic tools
Let P denote the set of all Borel probability measures on R. In the following, we explain main tools of free probability. Let C + := {z ∈ C : Im(z) > 0} and C − := {z ∈ C : Im(z) < 0}. For µ ∈ P, the Cauchy transform G µ : C + → C − is defined by and the reciprocal Cauchy transform F µ : C + → C + of µ ∈ P is defined by , z ∈ C + .
In this paper, we apply the Stieltjes inversion formula [A65, T00] for Boolean and monotone stable distributions. For any Borel probability measure µ, we can recover the distribution from its Cauchy transform: if µ does not have atoms at a, b, we have Im [a,b] G µ (x + iy)dx.
Especially, if G µ (z) extends to a continuous function on C + ∪ I for an open interval I ⊂ R, then the distribution µ has continuous derivative f µ = dµ/dx with respect to the Lebesgue measure dx on I, and we obtain f µ (x) by Im G µ (x + iy), x ∈ I.

Boolean case
In this paper, the maps z → z p and z → log z always denote the principal values for z ∈ C \ (−∞, 0]. Correspondingly arg(z) is defined in C \ (−∞, 0] so that it takes values in (−π, π).
Definition 1. Let b α,ρ be a boolean stable law [SW97] characterized by the following.
In the case α ∈ (0, 1) ∪ (1, 2], for simplicity we also use a parameter θ, instead of ρ, defined by (2.1) The probability measure b α,ρ is described as follows. Let Proposition 3. The Boolean stable distributions are as follows.
Proof. The case α = 2 is well known and we omit it.
If ρ = 1, then the computation for x > 0 is the same as (2.5). For x < 0, note that which has the unique zero at x = −u + (0). So G extends to a continuous function on . The weight of the atom at −u + (0) is equal to 1/a, and so we have (4).
π and u + . If ρ = 0, then the mode at u + is an atom.
π . If ρ = 1, then the mode at u − is an atom.
(ii) The case α ∈ (α 0 , 1). In this case 0 is still a mode of b α,ρ . From (2.8), we have that x → B α (x, θ) takes a local maximum in (0, ∞) ⇔ θ ∈ π 2 , απ and sin θ < α ⇔ θ ∈ (π − arcsin(α), απ], (2.9) and if this condition is satisfied, then the local maximum is attained at x = x + . By reflection, it holds that and if this condition is satisfied, the local maximum is attained at x = x − . The two conditions (2.9) and (2.10) cannot be satisfied for the same θ. Hence we have the conclusion (2).
(iv) The case α = 1, ρ = 1 2 . Note that the density takes 0 at x = 0. We have and then the remaining calculus is not difficult.
(b) The above definition does not respect the Bercovici-Pata bijection. If we hope to let m α,ρ correspond to b α,ρ regarding the monotone-Boolean Bercovici-Pata bijection, then we have to consider D α 1/α (m α,ρ ) which is the induced measure of m α,ρ by the map x → α 1/α x.
(c) All the above distributions are strictly stable. Non strictly stable distributions are not defined in the literature, and so we do not consider the non-symmetric case in α = 1.
Proposition 8. The strictly monotone stable distributions are as follows.
(4) If α ∈ (1, 2) and ρ ∈ (0, 1), then (5) If α ∈ (1, 2) and ρ = 1, then (6) If α = 2 and ρ ∈ [0, 1], then They are all absolutely continuous with respect to the Lebesgue measure. In the cases (α, ρ) ∈ (1, 2) × {0, 1} and α = 2, the density function diverges to infinity at the edge of the support, but in the other cases the density function is either continuous on R, or extends to a continuous function on R (if the support is not R). The density function is real analytic except at the edge of the support and at 0. The replacement ρ → 1 − ρ gives the reflection of the measure around x = 0.