On free infinite divisibility for classical Meixner distributions (Q2874354)

This paper treats free infinite divisibility of two distributions, namely, the Meixner distribution and the logistic distribution. A probability measure \(\nu\) on \({\mathbb R}\) is said to be freely infinitely divisible if, for any \(n \in {\mathbb N}\), there exists \(\nu_n\) such that NEWLINE\[NEWLINE \nu = \underbrace{ \nu_n \boxplus \nu_n \boxplus \cdots \boxplus \nu_n}_{\text{\(n\) times}}, \tag{1} NEWLINE\]NEWLINE where the free convolution \(\mu \boxplus \nu\) of probability measures \(\mu\) and \(\nu\) on \({\mathbb R}\) is the distribution of \(X + Y\) when \(X\) and \(Y\) are free self-adjoint random variables obeying the distributions \(\mu\) and \(\nu\), respectively, cf. [\textit{D. Voiculescu}, J. Funct. Anal. 66, 323--346 (1986; Zbl 0651.46063)]. As the first main result, the authors prove that symmetric Meixner distributions NEWLINE\[NEWLINE \rho_t( dx) = \frac{4^t}{ 2 \pi \Gamma( 2 t)} | \Gamma ( t + i x) |^2 dx, \quad x \in {\mathbb R}, \tag{2} NEWLINE\]NEWLINE are freely infinitely divisible for \(0 < t \leqslant \frac{1}{2}\), where \(\Gamma( z)\) is the gamma function. Especially for \(t = \frac{1}{2}\), the measure \(\rho_{1/2}\) coincides with the hyperbolic secant distribution \(\mu_1( dx)\) \(=\) \(\frac{1}{\cosh ( \pi x)} dx\), \(( x \in {\mathbb R})\), namely, the law of Lévy's stochastic area NEWLINE\[NEWLINE \frac{1}{2} \int_0^1 ( B_t^1 d B_t^2 - B_t^2 d B_t^1 ) \tag{3} NEWLINE\]NEWLINE with a standard two-dimensional Brownian motion \(( B_t^1, B_t^2)\). Secondly, the authors show that the logistic distribution NEWLINE\[NEWLINE \mu_2 ( dx) = \frac{ \pi}{ 2 \cosh^2 ( \pi x) } dx, \quad x \in {\mathbb R}, \tag{4} NEWLINE\]NEWLINE is freely infinitely divisible. Basic tools for proving free infinite divisibility of a probability measure \(\mu\) are the Cauchy transform \(G_{\mu}(z)\) and its reciprocal transform \(F_{\mu}(z)\) \(:=\) \(1 / G_{\mu} (z)\). They resort to a sufficient condition \(\mu\) \(\in\) \({\mathcal U}{\mathcal I}\) for the measure \(\mu\) to be freely infinitely divisible, named Arizmendi-Hasebe criterion. As a matter of fact, the class \({\mathcal U}{\mathcal I}\) is defined in terms of an analytical extension of \(F_{\mu}\) to a univalent map in a simply connected open set \(\Omega\) such that \({\mathbb C}^+\) \(\subset\) \(\Omega\) \(\subset\) \({\mathbb C}\), and the condition \(\mu \in {\mathcal U}{\mathcal I}\) yields the free infinite divisibility of \(\mu\). The proofs are technically due to \textit{O. Arizmendi} et al. [ALEA, Lat. Am. J. Probab. Math. Stat. 10, No. 1, 271--291 (2013; Zbl 1291.46060)], where the free infinite divisibility of the chi-square distribution \(\mu_3(dx)\) \(=\) \(\frac{1}{ \sqrt{ \pi x} } e^{-x} 1_{ [0, \infty)} (x) dx\) was proved.NEWLINENEWLINEThe basic characterization of the free convolution \(\mu \boxplus \nu\) via the Voiculescu transform \(\phi(z)\) was given by \textit{H. Bercovici} and \textit{D. Voiculescu} [Indiana Univ. Math. J. 42, No. 3, 733--773 (1993; Zbl 0806.46070)]. The free infinite divisibility of normal distributions was proved in [\textit{S. T. Belinschi} et al., Adv. Math. 226, No. 4, 3677--3698 (2011; Zbl 1226.46059)].