Free Perpetuities I: Existence, Subordination and Tail Asymptotics
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| Format: | Preprint |
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2025
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| _version_ | 1866913767793623040 |
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| author | Belinschi, Serban Kołodziejek, Bartosz Szpojankowski, Kamil |
| author_facet | Belinschi, Serban Kołodziejek, Bartosz Szpojankowski, Kamil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free Perpetuities I: Existence, Subordination and Tail Asymptotics Belinschi, Serban Kołodziejek, Bartosz Szpojankowski, Kamil Probability Operator Algebras Primary 46L54, Secondary 60E05 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. |
| title | Free Perpetuities I: Existence, Subordination and Tail Asymptotics |
| topic | Probability Operator Algebras Primary 46L54, Secondary 60E05 |
| url | https://arxiv.org/abs/2503.10319 |