[Paper Review] Densities of the Raney distributions

Wojciech Młotkowski, K. A. Penson+1 more|arXiv (Cornell University)|Nov 30, 2012

Random Matrices and ApplicationsMathematics18 references22 citations

TL;DR

This paper establishes that the Raney numbers $\binom{mp+r}{m}\frac{r}{mp+r}$ form a positive definite sequence for $p \geq 1$, $0 < r \leq p$, and proves the associated probability measures $\mu(p,r)$ are absolutely continuous with compact support on $[0,\infty)$ when $p > 1$ is rational. The density $W_{p,r}(x)$ is explicitly expressed via Meijer G-functions and generalized hypergeometric functions, with closed-form elementary expressions in special cases such as the multiplicative free square and square root of the Marchenko-Pastur law.

ABSTRACT

We prove that if $p\ge 1$ and $0&lt; r\le p$ then the sequence $\binom{mp+r}{m}\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more precisely, is the moment sequence of a probability measure $μ(p,r)$ with compact support contained in $[0,+\infty)$. This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at $x=2$. We show that if $p&gt;1$ is a rational number, $0

Motivation & Objective

• To prove that the Raney numbers $\binom{mp+r}{m}\frac{r}{mp+r}$ form a positive definite sequence for $p \geq 1$, $0 < r \leq p$, and thus define a unique probability measure $\mu(p,r)$ on $[0,\infty)$.
• To establish that $\mu(p,r)$ is absolutely continuous with a density $W_{p,r}(x)$ when $p > 1$ is rational.
• To derive an explicit representation of $W_{p,r}(x)$ in terms of Meijer G-functions and generalized hypergeometric functions.
• To identify cases where $W_{p,r}(x)$ reduces to elementary functions, including the multiplicative free square and square root of the Marchenko-Pastur distribution.
• To visualize the densities and analyze their support and sign behavior, particularly for fractional $p$ and varying $r$.

Proposed method

• Use of Mellin convolution to represent $\mu(p,r)$ as a product of modified beta measures.
• Application of the Mellin transform and properties of generalized hypergeometric functions to derive the density $W_{p,r}(x)$.
• Derivation of a closed-form expression for $W_{p,r}(x)$ using Meijer G-functions, valid for rational $p > 1$ and $0 < r \leq p$, involving hypergeometric functions with specific parameters.
• Reduction of the general hypergeometric expression to elementary functions in special cases via hypergeometric identity simplifications.
• Numerical and graphical analysis of the density functions for various rational $p$ and $r$, including support boundaries and sign behavior.
• Use of the monotonic convolution relation $\mu(p,r) \triangleright \mu(p+s,s) = \mu(p+s,r+s)$ to verify consistency with known results in free probability.

Experimental results

Research questions

• RQ1 For which values of $p \geq 1$ and $0 < r \leq p$ is the sequence $\binom{mp+r}{m}\frac{r}{mp+r}$ positive definite and thus a moment sequence of a probability measure $\mu(p,r)$?
• RQ2 What is the explicit form of the density $W_{p,r}(x)$ of $\mu(p,r)$ when $p > 1$ is rational?
• RQ3 In which cases does $W_{p,r}(x)$ reduce to an elementary function, and what are the corresponding closed-form expressions?
• RQ4 How do the densities $W_{p,r}(x)$ behave in terms of support, unimodality, and sign, particularly for fractional $p$ and varying $r$?
• RQ5 How do the Raney measures $\mu(p,r)$ relate to known distributions such as the Marchenko-Pastur and Wigner semicircle laws?

Key findings

• The sequence $\binom{mp+r}{m}\frac{r}{mp+r}$ is positive definite for $p \geq 1$, $0 < r \leq p$, and defines a unique probability measure $\mu(p,r)$ with compact support in $[0,\infty)$.
• For rational $p > 1$ and $0 < r \leq p$, the measure $\mu(p,r)$ is absolutely continuous and its density $W_{p,r}(x)$ is given by a combination of Meijer G-functions and generalized hypergeometric functions.
• In special cases such as $p=3$, $r=1,2,3$, the density $W_{3,r}(x)$ reduces to elementary functions involving $\sqrt{1-z}$ with $z = 4x/27$, yielding explicit closed forms.
• The multiplicative free square of the Marchenko-Pastur distribution ($\mu(3,1)$) corresponds to $W_{3,1}(x)$, which is expressed as a rational function of $\sqrt{1-z}$.
• The density $W_{p,1}(x)$ for $p=3/2$ and $r=1/2$ corresponds to the Bures distribution, and $W_{2,2}(x)$ is the Wigner semicircle law centered at $x=2$, confirming known results.
• Graphical analysis shows that for $r > p/2$, the densities $W_{p,r}(x)$ may develop negative parts, indicating non-positivity in certain parameter regimes.