Free Perpetuities I: Existence, Subordination and Tail Asymptotics
Serban Belinschi, Bartosz Kołodziejek, Kamil Szpojankowski
TL;DR
This work develops a comprehensive theory for the free analogue of affine perpetuities, solving X ≜ A^{1/2} X A^{1/2} + B under X being *-free from (A,B). It combines iterative series, free multiplicative convolution analysis, and subordination to establish existence in subcritical and critical regimes, with X unbounded when τ(A)=1, and to derive precise tail and moment behavior. Central contributions include a bilateral a.u. convergence result in the critical case, Tauberian-tail relations via subordination, and explicit examples using free Beta prime and inverse Marchenko–Pastur distributions that illustrate the Kesten-type tail phenomenon. The results illuminate how free probability qualitatively deviates from classical perpetuities, and offer a framework potentially applicable to matricial perpetuities and spectral distribution limits as dimension grows. Overall, the paper advances understanding of tail phenomena and fixed-point equations in noncommutative probability with concrete exemplifications and robust analytic machinery.
Abstract
We study the free analogue of the classical affine fixed-point (or perpetuity) equation \[ \mathbb{X} \stackrel{d}{=} \mathbb{A}^{1/2}\mathbb{X}\,\mathbb{A}^{1/2} + \mathbb{B}, \] where $\mathbb{X}$ is assumed to be $*$-free from the pair $(\mathbb{A},\mathbb{B})$, with $\mathbb{A}\ge 0$ and $\mathbb{B}=\mathbb{B}^*$. Our analysis covers both the subcritical regime, where $τ(\mathbb{A})<1$, and the critical case $τ(\mathbb{A})=1$, in which the solution $\mathbb{X}$ is necessarily unbounded. When $τ(\mathbb{A})=1$, we prove that the series defining $\mathbb{X}$ converges bilaterally almost uniformly (and almost uniformly under additional tail assumptions), while the perpetuity fails to have higher moments even if all moments of $\mathbb{A}$ and $\mathbb{B}$ exist. Our approach relies on a detailed study of the asymptotic behavior of moments under free multiplicative convolution, which reveals a markedly different behavior from the classical setting. By employing subordination techniques for non-commutative random variables, we derive precise asymptotic estimates for the tail of the distributions of $\mathbb{X}$ in both one-sided and symmetric cases. Interestingly, in the critical case, the free perpetuity exhibits a power-law tail behavior that mirrors the phenomenon observed in the celebrated Kesten's theorem.