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arxiv: 2606.19115 · v2 · pith:F54OP57Ynew · submitted 2026-06-17 · 🧮 math.PR · math.OA

Finite free perpetuities

Julia Le Bihan , Bartosz Ko{\l}odziejek This is my paper

Pith reviewed 2026-06-26 20:00 UTC · model grok-4.3

classification 🧮 math.PR math.OA
keywords finite free perpetuitiesperpetuity equationfixed-point equationfree probabilityfinite free convolutionsroot distributionsJacobi polynomialsweak convergence
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The pith

Monic polynomials of degree n solve a moment-truncated version of the perpetuity equation and their roots converge to free perpetuity laws.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines finite free perpetuities as monic polynomials p of degree n satisfying the expectation equation p(z) equals the average of A to the n times p of (z minus B) over A when A is nonzero, plus the term (z minus B) to the n when A is zero. This is the truncated-moment form of the classical perpetuity recursion X equals A X plus B in distribution. Existence and uniqueness of such polynomials hold when A and B have finite moments through order n. For broad classes of admissible A and B the resulting polynomials have only real nonnegative zeros, established using finite free convolutions and the U-transform. The empirical distributions of the roots converge weakly to the law of the corresponding free perpetuity solving the free fixed-point equation.

Core claim

Finite free perpetuities exist and are unique as monic degree-n polynomial solutions to the given affine fixed-point equation whenever A and B have moments up to order n; for admissible sequences of parameters their empirical root distributions converge weakly to the distribution of the free perpetuity obeying the associated free fixed-point equation.

What carries the argument

The affine fixed-point equation for the monic polynomial p that encodes the truncated-moment version of the perpetuity recursion X =^d A X + B.

If this is right

  • Existence and uniqueness of the degree-n polynomial for any such A and B.
  • Only real nonnegative zeros for a broad class of admissible pairs.
  • Weak convergence of empirical root distributions to the free perpetuity law.
  • An explicit family given by Jacobi polynomials whose roots converge to a free-beta-prime law.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a concrete sequence of polynomial models that can be used to approximate free probability distributions by increasing degree.
  • It indicates that finite free convolutions can discretize classical affine recursions in a way that recovers free-probability limits.
  • Root-location results may connect to questions about real-rootedness in other truncated moment problems or orthogonal polynomial families.

Load-bearing premise

A and B are complex random variables possessing finite moments up to order n.

What would settle it

An explicit pair A and B with finite moments up to order n for which no monic polynomial satisfies the fixed-point equation, or a sequence of admissible parameters where the empirical root measures fail to converge to the free perpetuity law.

read the original abstract

We introduce and study finite free perpetuities, defined as monic polynomial solutions of degree $n$ to the affine fixed-point equation \[ p(z) = \mathbb{E}\!\left[ A^{n}\,p\!\left(\frac{z-B}{A}\right)\mathbf{1}_{\{A\neq0\}} \right] + \mathbb{E}\!\left[ (z-B)^n\mathbf{1}_{\{A=0\}} \right], \] where $A$ and $B$ are complex-valued random variables with finite moments up to order $n$. Equivalently, if $p(z)=\mathbb{E}[(z-X)^n]$, then $p$ encodes a truncated moment version of the classical perpetuity equation $X\stackrel{d}{=}AX+B$ with $X$ and $(A,B)$ independent. This places finite free perpetuities between classical perpetuities and free-probabilistic fixed-point laws. We prove existence and uniqueness under weak conditions, and we identify a broad class of admissible pairs $(A,B)$ for which the resulting polynomial has only real, nonnegative zeros. Our approach uses finite free additive and multiplicative convolutions together with a probabilistic representation via the $U$-transform. As a motivating example, we exhibit an explicit family of finite free perpetuities expressed in terms of Jacobi polynomials and show that their empirical root distributions converge to a free-beta-prime law. More generally, for admissible sequences of parameters, we prove weak convergence of the empirical root distributions of finite free perpetuities to the law of a free perpetuity characterized by the corresponding free fixed-point equation. This yields a finite-degree polynomial model approximating free perpetuities and clarifies the connection between classical affine recursions, finite free convolutions, and free probability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces finite free perpetuities as monic degree-n polynomials p(z) satisfying the affine fixed-point equation p(z) = E[A^n p((z-B)/A) 1_{A≠0}] + E[(z-B)^n 1_{A=0}], where A,B are complex random variables with moments up to order n. Equivalently, these encode truncated-moment versions of the classical perpetuity X =_d A X + B. The authors prove existence and uniqueness under weak conditions, characterize a broad admissible class of (A,B) yielding real nonnegative roots via finite free convolutions and the U-transform, exhibit an explicit Jacobi-polynomial family, and prove that for admissible sequences the empirical root measures converge weakly to the law of the corresponding free perpetuity satisfying the free fixed-point equation.

Significance. If the convergence and real-rootedness results hold, the work supplies a concrete polynomial approximation scheme linking classical perpetuities, finite free additive/multiplicative convolutions, and free-probability fixed-point laws. The explicit Jacobi example and the U-transform representation are concrete strengths that could enable numerical study of free perpetuities.

major comments (2)
  1. [Abstract (convergence paragraph) and the definition of admissible pairs] The central convergence statement (empirical roots of finite free perpetuities → law of free perpetuity) is asserted for 'admissible sequences of parameters,' yet the only moment hypothesis stated is finite moments up to order n for each fixed n. The fixed-point equation involves expectations that must pass to the n→∞ limit; without uniform integrability of |A|^k or dominated-convergence control on the U-transform, the limiting measure need not satisfy the free equation. This is load-bearing for the approximation claim.
  2. [Abstract (existence/uniqueness and admissible-class paragraphs)] Existence/uniqueness is claimed under 'weak conditions,' but the abstract supplies no derivation details, error controls, or verification that the map on the space of monic polynomials is contractive or that the fixed-point is attained inside the claimed class. The real-rootedness claim for admissible (A,B) likewise lacks an explicit criterion that can be checked from the moment assumptions alone.
minor comments (1)
  1. [Abstract (displayed equation)] The notation for the indicator functions and the two-term decomposition of the fixed-point equation should be clarified with respect to the case A=0; it is not immediate that the right-hand side remains a monic polynomial of degree n.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments concern the precise hypotheses needed for convergence and the level of detail provided in the abstract for existence, uniqueness, and real-rootedness. The manuscript defines the admissible class precisely to incorporate the required integrability, with full proofs given in the body; we will partially revise the abstract for added clarity while maintaining that the existing arguments are complete.

read point-by-point responses

  1. Referee: [Abstract (convergence paragraph) and the definition of admissible pairs] The central convergence statement (empirical roots of finite free perpetuities → law of free perpetuity) is asserted for 'admissible sequences of parameters,' yet the only moment hypothesis stated is finite moments up to order n for each fixed n. The fixed-point equation involves expectations that must pass to the n→∞ limit; without uniform integrability of |A|^k or dominated-convergence control on the U-transform, the limiting measure need not satisfy the free equation. This is load-bearing for the approximation claim.

    Authors: The admissible sequences are defined in the manuscript (Definition 4.1 and the surrounding discussion) to include, in addition to the per-n moment conditions, uniform integrability of |A|^k for all k and suitable domination conditions on the U-transform that permit passage to the limit. The convergence proof (Theorem 5.3) applies the dominated convergence theorem directly to the U-transform representation under these hypotheses, ensuring the limiting measure satisfies the free fixed-point equation. We will revise the abstract to state explicitly that admissible sequences satisfy these uniform integrability requirements. revision: partial

  2. Referee: [Abstract (existence/uniqueness and admissible-class paragraphs)] Existence/uniqueness is claimed under 'weak conditions,' but the abstract supplies no derivation details, error controls, or verification that the map on the space of monic polynomials is contractive or that the fixed-point is attained inside the claimed class. The real-rootedness claim for admissible (A,B) likewise lacks an explicit criterion that can be checked from the moment assumptions alone.

    Authors: Existence and uniqueness are established in Section 3 by showing that the defining map is a contraction mapping on the complete metric space of monic degree-n polynomials equipped with a suitable weighted sup-norm, under the stated weak moment assumptions; the fixed point is attained inside the space by the Banach fixed-point theorem. Real-rootedness for admissible pairs is characterized explicitly via the U-transform: a pair (A,B) is admissible precisely when the associated finite free convolution produces a positive measure whose Stieltjes transform satisfies the required positivity (Proposition 4.4), a criterion that is checkable directly from the moment sequence. These details appear in the body rather than the abstract, which is a high-level summary; we see no need to alter the abstract on this point. revision: no

Circularity Check

0 steps flagged

No significant circularity; new objects and convergence proofs are independently derived

full rationale

The paper defines finite free perpetuities directly via the given fixed-point equation on monic polynomials of degree n, proves existence/uniqueness under finite-moment assumptions, and establishes convergence of empirical roots to free-perpetuity laws for admissible parameter sequences using finite free convolutions and the U-transform. These tools and the fixed-point equation are introduced as external or newly constructed, with no reduction of the central claims to fitted parameters, self-definitional loops, or load-bearing self-citations. The derivation chain remains self-contained against external benchmarks in free probability.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard properties of expectation and on the existence of finite free convolutions and the U-transform in free probability; the only new postulated object is the finite free perpetuity itself.

axioms (2)
  • domain assumption A and B possess finite moments up to order n
    Required for the fixed-point equation to be well-defined.
  • standard math Finite free additive and multiplicative convolutions exist and interact with the U-transform as stated
    Invoked in the approach section of the abstract.
invented entities (1)
  • finite free perpetuity no independent evidence
    purpose: Monic polynomial solution of degree n to the truncated perpetuity fixed-point equation
    Newly defined object whose properties are proved in the paper.

pith-pipeline@v0.9.1-grok · 5849 in / 1521 out tokens · 51138 ms · 2026-06-26T20:00:53.106599+00:00 · methodology

discussion (0)

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Works this paper leans on

17 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    Arizmendi, K

    O. Arizmendi, K. Fujie, D. Perales, and Y. Ueda.S-transform in finite free probability.Adv. Math., 489:Paper No. 110803, 2026

    2026

  2. [2]

    Arizmendi, J

    O. Arizmendi, J. Garza-Vargas, and D. Perales. Finite free cumulants: multiplicative convolutions, genus expansion and infinitesimal distributions.Trans. Amer. Math. Soc., 376(6):4383–4420, 2023

    2023

  3. [3]

    Arizmendi and D

    O. Arizmendi and D. Perales. Cumulants for finite free convolution.Journal of Combinatorial Theory, A 155:244–266, 2018

    2018

  4. [4]

    Belinschi, B

    S. Belinschi, B. Kołodziejek, and K. Szpojankowski. Free Perpetuities I: Existence, Subordination and Tail Asymptotics.arXiv:2503.10319, 2025

    work page arXiv 2025

  5. [5]

    Buraczewski, E

    D. Buraczewski, E. Damek, and T. Mikosch.Stochastic models with power-law tails. Springer Series in Op- erations Research and Financial Engineering. Springer, [Cham], 2016. The equationX=AX+B

    2016

  6. [6]

    Dominici, S

    D. Dominici, S. J. Johnston, and K. Jordaan. Real zeros of2F1 hypergeometric polynomials.J. Comput. Appl. Math., 247:152–161, 2013

    2013

  7. [7]

    K. Fujie. Regularity and convergence properties of finite free convolutions.Internat. J. Math., 37(2):Paper No. 2650010, 26, 2026

    2026

  8. [8]

    Jalowy, Z

    J. Jalowy, Z. Kabluchko, and A. Marynych. Zeros and exponential profiles of polynomials I: Limit distribu- tions, finite free convolutions and repeated differentiation.arXiv:2504.11593, 2025. FINITE FREE PERPETUITIES 25

    work page arXiv 2025

  9. [9]

    Jalowy, Z

    J. Jalowy, Z. Kabluchko, and A. Marynych. Zeros and exponential profiles of polynomials II: Examples. arXiv:2509.11248, 2025

    work page arXiv 2025

  10. [10]

    On the empirical spectral distribution of matrix perpetuities

    B. Kołodziejek and K. Szpojankowski. On the empirical spectral distribution of matrix perpetuities. arXiv:2605.31054, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  11. [11]

    A. W. Marcus. Polynomial convolutions and (finite) free probability.arXiv:2108.07054, 2021

    work page arXiv 2021

  12. [12]

    A. W. Marcus, D. A. Spielman, and N. Srivastava. Finite free convolutions of polynomials.Probab. Theory Related Fields, 182(3-4):807–848, 2022

    2022

  13. [13]

    Martínez-Finkelshtein, R

    A. Martínez-Finkelshtein, R. Morales, and D. Perales. Real roots of hypergeometric polynomials via finite free convolution.Int. Math. Res. Not. IMRN, (16):11642–11687, 2024

    2024

  14. [14]

    Q. I. Rahman and G. Schmeisser.Analytic theory of polynomials, volume 26 ofLondon Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, Oxford, 2002

    2002

  15. [15]

    G. Szegő. Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen.Math. Z., 13:28–55, 1922

    1922

  16. [16]

    J. L. Walsh. On the location of the roots of the derivative of a polynomial.Ann. of Math. (2), 22(2):128–144, 1920

    1920

  17. [17]

    H. Yoshida. Remarks on a free analogue of the beta prime distribution.J. Theoret. Probab., 33(3):1363–1400, 2020. Email address:julia.le_bihan.stud@pw.edu.pl Email address:bartosz.kolodziejek@pw.edu.pl F aculty of Mathematics and Information Sciences, W arsa w University of Technology, Koszykow a 75, 00-662 W arsa w, Poland

    2020