KANTER RANDOM VARIABLE AND POSITIVE FREE STABLE DISTRIBUTIONS

We derive the representative Bernstein measure of the density of ( X α ) − α/ ( 1 − α ) ,0 < α < 1, where X α is a positive stable random variable, as a Fox-H function. Up to a factor, this measure describes the law of some function a α of a uniform random variable U on ( 0, π ) . The distribution function of a 1 − α ( U ) is then expressed through a H-function and is used to describe more explicitly the density of the analogue of X α in the setting of free probability theory. Moreover, this density is shown to converge to a function with inﬁnite mass when α → 0 + , in contrast to the classical setting where X α is known to converge weakly to the inverse of an exponential random variable. Analytic evidences of the occurence of a α in both the classical and the free settings conclude the exposition.

The non trivial positivity of this series follows for instance from the integral representation displayed p. 74 in [19], as most of analytic properties of stable distributions do from their numerous path-integral representations. Amazing and non trivial as well is the complete monotonicity of the density of (X α ) −r , r ≥ α/(1 − α) (in particular its infinite divisibility, [15]). Indeed, there exist a standard exponential random variable L and an infinitely divisible random variable V α,r , both variables being independent, such that ( [13]): For the special value r = α/(1 − α), the following very precise description of V α,r traces back implicitly [5] (see also Ch.II in [10]) and is due to M. Kanter ([11] p. 703) where U is a uniform random variable on (0, π) and (1−α) , c α (u) := sin(αu) sin(u) .
Hence, the injectivity of the Laplace transform entails and we refer below to a α (U) as the Kanter random variable. Since a α is strictly increasing ( [11] p. 704), let a −1 α (u) = π (a α (U) ≤ u), u ≥ a α (0) = (1 − α)α α/(1−α) , be its inverse function. Mysteriously, a −1 1−α appears in the expression of the density of the analogue of X α , 0 < α < 1 in the setting of Voiculescu's free probability theory. The analogy comes from Voiculescu's free additive convolution giving rise to the so-called free stable distributions ( [1], [2], [3]). The latter distributions form a two-parameters family (up to a drift and a scaling parameters). Moreover, keeping the notations from [2], the set {0 < α < 1, ρ = 1} corresponds exactly to distributions supported in (0, ∞), that we shall call in short positive free distributions. More precisely, their densities may be written in the following form ( [2] p. 1053) for x ≥ a 1−α (0). For this reason, we think that it is reasonable to have an insight on the mysterious occurence of the Kanter random variable in both probability theories. It is also of some benefit to seek an expression of the density of a α (U) as a special function, and more generally of e V α,r , since then (1) would be more explicit. The easiest way to address the second problem is to invert the Mellin transform of the density of e −V α,r given by ( [13] p. 292) (e −sV α,r ) = Γ((rs/α) + 1) However, Mellin's inversion formula 1 2iπ y +i∞ −i∞ Γ((rs/α) + 1) applies for any r > α/(1 − α) (take ℜ(s) = 0 as a path of integration) but does not when r = α/(1−α). The latter fact may be seen from the following estimate of |Γ(a + i b)|, a, b ∈ , |b| → ∞ displayed in formula 1.24. p. 4 from [9]: so that the absolute value of the integrand in (2) is equivalent to |s| −1/2 (up to a constant). Note that when r > α/ (1−α), the integral displayed in (2) The first result of this paper states then that in the pathological case r = α/(1 − α), the density of a α (U) is still a H-function yet represented through a different path integral than the one used in (2) (Hankel path). Using Cauchy's residue Theorem, our second result expresses a −1 1−α through another H-function thereby describes more explicitly the density (1). Next, we investigate the limiting behavior of the density appearing in (1) aiming to prove a free analogue of Cressie's result on the weak convergence of stable laws when α → 0 + ( [6]). In the last and likely the most important part of this paper, we supply analytic arguments trying to describe the occurence of a α in both probabilistic settings (classical and free). Loosely speaking, the inverse formula for the Fourier transform of the distribution of X α together with admissible deformations of the paths of integration lead to the so-called Voiculescu's transform of the distribution of its free analogue. Nevertheless, we think that a group theoretical argument should exist, hidden at this point. For the sake of completeness, some basic facts on the H-function are collected in the next section. A good reference on this topic is the monograph [9].

On the Fox H-function
The Fox H-function is defined as a Mellin-Barnes integral Here, we take the principal determination of the power function (although this is not strictly necessary) and furthermore agree that an empty product is equal to one. The integral is taken over a suitable contour L separating the poles of both products of the meromorphic (Gamma) functions in the numerator of Θ ( [9] p. 3,4). The choice of L and of the domain of convergence of the Mellin-Barnes integrals defining the H-function for each contour depend on the parameters Moreover, these integrals are evaluated by means of Cauchy's residue Theorem applied to truncated contours together with asymptotic estimates at infinity of the integrand. As a matter of fact, a contour may be deformed into another one provided that the deformation is admissible ( [7], [17], see also CH.V. in [8] for a similar discussion on Meijer's G-function which coincides with the H-function when Eight cases are discussed in [9] p. 4. We only make use of five of them in the sequel to which we refer whenever needed. Set then the H-function is well-defined for complex numbers lying on the circle |z| = ∆.

v)
If then the H-function exists for all z = 0 such that | arg(z)| < πΩ/2 and L is the infinite semi-circle γ − i∞, γ + i∞ for a suitable real γ.

Remark 1.
In the sequel, the complex variable z will take only strictly positive values, thereby s → z −s is defined through the real-valued logarithm function.

The density of a α (U)
Let h α denote the density of a α (U) supported by (a α (0), ∞), then the density of Hence, one needs to derive a Laplace-type integral representation of the LHS of this equality. The issue of this derivation is the following statement.

Proposition 1. The density h α of the Kanter variable is expressed as
Proof. Using the mirror formula satisfied by the Euler Gamma function ( [8], p. 3), According to A.6 p. 218 in [9], the latter density is expressed through a H-function 2 : for all x > 0, which is well defined for all z = 0 in the complex plane by the virtue of (i) since q ≥ 1, µ = 2 − 2α > 0. According to (v) and using the Gamma integral, one has for γ > 0 In order to change the order of integration, one needs to check that Performing an integration with respect to y, it amounts to check which follows again from the estimate of |Γ(a + i b)|, |b| → ∞ (see (3)). Thus, Fubini's Theorem yields Note that the H-function displayed in (4)

/ j) (U) is distributed as a product of independent Beta variables since the H-function involved in h 1−(1/ j) reduces to a Meijer function ([9] Ch.IV).
This description is not surprising since X 1/ j is distributed as a product of ( j − 1) independent Gamma variables ( [18], see also [16] for a more larger set of special indices).

Positive free stable distributions
Once h α is written down, the distribution function of a α (U) is expressed below through a Hfunction too. The density (1) is then described more explicitly and we shall investigate the limiting behavior, when α → 0 + , of its image under the map x → x α .

Distribution function of a 1−α (U)
Before stating our second result, let us consider the case α = 1/2. Then For general α ∈ (0, 1), one derives the following conclusion.
Proof. Assuming for a while that Fubini's Theorem applies, the integral representation is defined on the whole real line by the virtue of (ii),(iii) and (iv), it follows that .
Finally, the continuity of the H-function entails for any x < α(1 − α) (1−α)/α by the virtue of (iii). Coming back to the validity of Fubini's Theorem assumed at the beginning of the proof, one needs to ensure, after integrating with respect to the variable y, the convergence of the integral But, this holds true since the estimate of Γ(a + i b) for large a (formula 1.23 p. 4. in [9]): shows that the integrand is equivalent to C |ℜ(s)| 3/2 e −(α log(α)+(1−α) log(1−α))ℜ(s) , |ℜ(s)| → ∞, for some constant C = C(ℑ(s)) depending only on ℑ(s). The proof of the proposition is therefore complete.

Remarks 1. 1/Elementary variables changes give
. [2] p. 1055 allows to derive a more explicit description of the density of a free stable distribution of index 1 < α < 2 and asymmetry coefficient ρ = 0.

Behavior for small indices
Motivated by a limiting result for general stable variables as α → 0 + due to N. Cressie ([6], see also [4]), we now investigate the limiting behavior of a α (U) and of the image of the density displayed in (1) under the map x → x α as α → 0 + . To proceed, recall that if Y (α, β, γ, c), 0 < α ≤ 2, |β| ≤ 1, γ ∈ , c > 0 is a stable random variable, then in distribution, where L is an exponential random variable with mean 1/c (when c = 0, Y (α, β, γ, 0) degenerates). Without loss of generality, we may choose c = 1. Noting that this limiting result does not depend on the asymmetry parameter β and that the special set of parameters {0 < α < 1, β = 1, γ = 0, c = 1} corresponds to positive stable random variables, Cressie's result reduces to When combined with Kanter's representation , the latter limiting result shows that a α (U) has to be degenerate when α tends to 0 + . In the free setting, the picture is quite different.
Proof. Note first that the LHS is the image of the density displayed in (1) under the map x → x α and that it is supported in {x > α α (1 − α) 1−α }. Now, according to Appendix 6 of [9] p. 218, one expands Next, the mirror formula together with Lebesgue's convergence Theorem yield the limit
It would be interesting to perform a 'holomorphic' change of variable z → 1/z which maps C α onto ∂ Ω 1−α , since it may lead to a relation between the densities of positive stable variables in both probabilistic settings. Note that if is the Cayley map satisfying ψ(u) = 1/(ψ(u)) then ψ −1 α and ψ −1 ∂ Ω 1−α are complex conjugate.