On the empirical spectral distribution of matrix perpetuities

Bartosz Ko{\l}odziejek; Kamil Szpojankowski

On the empirical spectral distribution of matrix perpetuities

T0 review · 0 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Matrix perpetuities have power-law eigenvalue tails governed by the largest eigenvalue.

desk verdict Solid new results on ESDs of matrix perpetuities under orthogonal invariance; the spectral Kesten theorem holds up, with the orthogonal-invariance restriction being the main scope limit. read the letter →

arxiv 2605.31054 v3 pith:66UVSMD6 submitted 2026-05-29 math.PR

Bartosz Ko{l}odziejek , Kamil Szpojankowski This is my paper · ORCID

classification math.PR MSC 60B2060H2546L5460G70

keywords matrixperpetuityempiricalspectraldistributiontailasymptoticsKestentheoremorthogonalinvariancefreebetaprimerandommatrices

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a spectral analogue of the classical Kesten theorem for matrix perpetuities, solutions of X = A X A^T + B. Under orthogonal invariance and positive semidefiniteness, the expected empirical spectral distribution has a power-law tail, that tail is asymptotically equal to (1/N) times the tail of the largest eigenvalue, and the prefactor is explicitly expressed through the tail of the scalar perpetuity satisfied by the top-left entry. The central reduction is that principal submatrices of a matrix perpetuity are themselves lower-dimensional perpetuities, so the whole spectral tail is controlled by a one-dimensional recursion. This matters because such recursions model heavy-tailed stochastic algorithms, and the result gives a parameter-free prediction of how multiplicative noise and additive perturbations shape the eigenvalue distribution. In the subcritical regime, the expected ESD converges weakly to the corresponding free perpetuity as dimension grows.

What carries the argument

The engine is the principal-compression identity for symmetric multiplicative convolution: if B is orthogonally invariant, then for every k the k×k leading principal submatrix of A B A^T has the same law as ((AA^T)^[k])^{1/2} B^[k] ((AA^T)^[k])^{1/2}. Iterating along the product M_n = Π_n Π_n^T shows that principal submatrices of the perpetuity itself solve matrix perpetuities of smaller dimension, and for k=1 reduces to the scalar perpetuity X_11 = (AA^T)_11 X_11 + B_11. This reduction also identifies the top Lyapunov exponent as E[log(AA^T)_11], replacing the usual spectral-radius condition by the scalar moment condition E[A_11^η]=1.

What would settle it

Simulate a matrix Beta prime perpetuity with parameters near the critical case, where E[det(A^[2])^{η/2}] approaches 1, and compare the empirical eigenvalue tail with the formula; specifically, check whether P(λ_2>t)/P(λ_1>t) tends to 0 and whether the tail prefactor matches √π Γ(η+N/2)/(Γ(N/2)Γ(η+1/2)) times the X_11 tail. A mismatch there would pinpoint a failure of the principal-compression reduction.

Watch

Extended reading notes

Core claim

Theorem 5.1: for N≥2, if A,B are a.s. positive semidefinite and (A,B) orthogonally invariant, and η>0 solves E[A_11^η]=1 with moment and non-arithmeticity conditions, the unique solution X satisfies μ_X(t,∞)∼(1/N)P(λ_max(X)>t)∼(1/N)[√π Γ(η+N/2)/(Γ(N/2)Γ(η+1/2))]P(X_11>t), where X_11 solves X_11=(AA^T)_11 X_11 + B_11 and its tail has the explicit constant E[(A_11 X_11+B_11)^η − (A_11 X_11)^η]/(η E[A_11^η log A_11]). The expected eigenvalue spectrum is then a power law of index η, dominated in its tail by the largest eigenvalue, with all constants explicit in terms of the (1,1) entries and the dimension. In the subcritical regime τ(A)<1, the expected ESD converges weakly to the free perpetuity

Load-bearing premise

The load-bearing assumption is orthogonal invariance of the pair (A,B) together with A,B ≥ 0; if the update law is not invariant under conjugate rotations, the principal-submatrix reduction to a scalar perpetuity fails and the tail index is no longer E[A_11^η]=1.

Editorial extensions

If this is right

The expected empirical spectral distribution of a matrix perpetuity has a power-law tail t^{-η}, with η determined by the one-dimensional condition E[A_11^η]=1.
The tail is asymptotically (1/N) times the tail of the largest eigenvalue; the probability that the second eigenvalue exceeds t is negligible compared with the largest eigenvalue tail.
The prefactor connecting the spectral tail to P(X_11>t) is explicit: √π Γ(η+N/2)/(Γ(N/2)Γ(η+1/2)), times the scalar tail, and the scalar tail constant is given in closed form.
In the subcritical regime τ(A)<1, the expected ESD converges weakly to the free perpetuity distribution, so the bulk spectrum is asymptotically free while the extreme eigenvalues carry the heavy tail.
For matrix Beta prime perpetuities, the tail index is explicit: η_1 = β − (N−1)/2, with an explicit multiplicative constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The compression mechanism is not specific to real symmetric matrices; the same triangular-factor argument should hold for complex Hermitian models under unitary invariance, giving the same reduction to a scalar perpetuity for the (1,1) entry.
This suggests a concrete diagnostic for heavy-tailed stochastic optimization: fit the top-left entry of the update matrix, solve E[A_11^η]=1, and compare the predicted power-law exponent with the empirical eigenvalue tails.
The tail-bulk separation proved here — a free bulk plus Kesten-type extremes — is probably a general phenomenon for orthogonally invariant affine random-matrix recursions, and may hold beyond the perpetuity equation.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuine advance and the main theorem checks out. The paper opens the spectral analysis of matrix perpetuities X = A X A^T + B, which nobody has done, and the two headline results are a finite-dimensional spectral Kesten theorem and a weak convergence to free perpetuities in the subcritical regime.

What is actually new: the principal-submatrix compression identity (Cor 4.5) is clean and appears new; it lets the authors prove that principal submatrices of a matrix perpetuity are themselves perpetuities of the same form. That is what reduces the (1,1) entry to a scalar perpetuity X11 = A11 X11 + B11, and Theorem 5.1 then gives precise power-law tails for the expected ESD: the tail is governed by the largest eigenvalue, and the prefactor is explicit in terms of E[U^{2η}] and the Goldie constant. The proof is coherent. The moment lemmas are real work, the use of Goldie's theorem is legitimate, and the Converse Breiman step is carefully checked with the non-cancellation condition. I don't see a circularity problem: the tail theorem relies on classical vector perpetuity results, and the self-citation to [5] is ordinary reliance for the existence of the free limit.

Soft spots. The orthogonal invariance plus positive semidefiniteness assumption in Theorem 5.1 is genuinely restrictive; the whole compression argument depends on it, and it is not just a technical convenience. The paper is transparent about that, but it does mean the result covers a specific symmetry class of models. The weak convergence theorem is only subcritical (τ(A)<1), and the authors note the critical case is open. The least standard step, as the stress test says, is the Converse Breiman inversion that turns the tail of U^2 λ1(X) into the tail of λ1(X); that relies on Jacobsen–Mikosch–Rosiński–Samorodnitsky and the non-vanishing condition. On reading, the application is legitimate, but it is the one place I'd want a referee to look closely.

The matrix Beta prime example is a nice illustration, with explicit limiting spectral densities via a Coulomb gas argument. Not everything there is new, but it is useful.

Bottom line: this deserves a serious referee. The core theorem is likely correct, the exposition is careful, and the restrictions are stated openly. I would accept it with the usual request for clarity on the Converse Breiman step and for an explicit statement of how restrictive the orthogonal invariance is.

Referee Report

0 major / 4 minor

Summary. The paper studies finite-dimensional matrix perpetuities, i.e. solutions of the affine fixed-point equation X = A X A^⊤ + B (or, in the positive semidefinite case, X = A^{1/2} X A^{1/2} + B), with emphasis on the expected empirical spectral distribution. Existence and uniqueness are first obtained by vectorization and classical vector-perpetuity theory under a negative top Lyapunov exponent. Under orthogonal invariance and positive semidefiniteness, the authors prove a compression identity: principal submatrices of a matrix perpetuity are again perpetuities driven by the corresponding compressed coefficients. This yields the main theorem (Theorem 5.1): under E[A_{11}^η] = 1 and regularity conditions, μ_X(t,∞) is asymptotically equal to (1/N) P(λ_max(X) > t), and to an explicit constant times P(X_{11} > t), where X_{11} is the scalar perpetuity A_{11}X_{11}+B_{11} and the tail constant is given by Goldie's theorem. In the subcritical regime τ(A)<1, the expected ESD is shown to converge weakly to the law of the corresponding free perpetuity. The results are illustrated by matrix Beta prime perpetuities, where explicit limiting spectral distributions are available.

Significance. If correct, Theorem 5.1 is a finite-dimensional spectral Kesten theorem: it gives precise power-law tail asymptotics for the expected empirical spectral distribution of a matrix perpetuity, identifies the largest eigenvalue as the sole contributor to the tail, and connects the tail constant to the classical scalar perpetuity. The compression identity for principal submatrices under orthogonal invariance is a new structural tool, and the weak convergence to free perpetuities in the subcritical regime is a meaningful bridge between random matrix theory and free probability. The proof is detailed and internally coherent: the scalar reduction is justified by Lemma 4.13, the moment bounds on λ1λ2 are supplied by Lemmas 5.4–5.7, the tail of X_{11} is handled by Goldie's theorem, and the Converse Breiman step is explicitly checked. The paper is honest about the orthogonal-invariance and positive-semidefiniteness restrictions, and the matrix Beta prime example provides explicit, falsifiable constants.

minor comments (4)

[Theorem 5.1 / Section 5] The theorem is stated for the recursion X = A^{1/2} X A^{1/2} + B, while Sections 3–4 and the auxiliary lemmas in Section 5.1 are written for X = A X A^T + B. The proof is coherent only after replacing the coefficient by C = A^{1/2} throughout, so that C C^T = A. This substitution is never stated explicitly. Please add a sentence at the beginning of Section 5 clarifying that all lemmas from Section 4 are applied with C = A^{1/2}; equivalently, the notation in Theorem 5.1 refers to the invariant matrix C C^T.
[Lemma 5.7, proof around Eq. (5.20)–(5.22)] The application of Lemma 5.5 to the m-step recursion X = Π_m X Π_m^T + C_m is compressed. In particular, the condition h1(p)<1 for the original coefficient implies the corresponding condition h_{Π_m}(p)=h1(p)^m<1 for the coefficient Π_m; this is not spelled out. A short justification would make the argument easier to verify.
[Theorem 5.1, final step (Converse Breiman)] The Converse Breiman step is the least standard ingredient. The non-cancellation condition E[U^{2η+2iθ}]≠0 is checked by the displayed Gamma-function formula, and the application appears valid. For the reader, please state explicitly the exact form of [20, Theorem 4.2] used, and note that it supplies both the regular variation of P(λ1(X)>t) and the constant 1/E[U^{2η}], not merely the asymptotic comparability.
[Section 6, end] The remark that the critical regime τ(A)=1 is not covered is useful. It might be worth adding a sentence that the tail exponents of the finite-dimensional perpetuities are not expected to match the free tail exponent, since the paper already notes the discontinuity of tail behavior under weak convergence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral Kesten theorem is derived from external classical results, not from self-referential fits.

full rationale

I traced the derivation chain. Existence (Theorem 3.1) is obtained by vectorization and cited to the classical vector-perpetuity theorem [8, Thm 4.1.4], not to the paper's own conclusions. The principal-submatrix compression identity (Corollary 4.5 and Lemma 4.13) is proved from the measurable polar decomposition (Lemma 4.2 / Appendix C) and Cholesky structure; it is a structural identity, not an input-output tautology. The main tail theorem (Theorem 5.1) rests on independent external results: Goldie's scalar perpetuity tail theorem [15, Thm 4.1] for X11; the moment lemmas 5.4-5.8, proved by interlacing, elementary symmetric polynomials, and submultiplicativity; and the Converse Breiman lemma [20, Thm 4.2], whose non-cancellation condition is checked explicitly via non-vanishing Gamma moments. The condition E[A11^eta]=1 is the classical Kesten parameter, and the constant in (5.2) is Goldie's constant; it is not a fitted value renamed as a prediction. Self-citations to [5] in Theorem 2.4 and Section 6 supply existence and uniqueness of the free perpetuity limit. These are prior theorems with stated assumptions, not data-dependent fits, and Theorem 6.2 proves convergence to that limiting object rather than assuming convergence. The paper itself flags genuine scope limitations -- orthogonal invariance / positive semidefiniteness in Theorem 5.1 and the exclusion of the critical case tau(A)=1 in Theorem 6.2 -- and these are restrictions of the model, not evidence of circularity. No load-bearing step reduces, by construction or by self-citation, to its own input.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central theorems introduce no fitted numerical constants: the tail index η is determined by the moment equation E[A_11^η]=1, not chosen from data. The main additional load-bearing input beyond standard probability results is the orthogonal-invariance/PSD domain assumption, plus the technical Goldie/non-arithmetic conditions. The weak-convergence theorem also leans on the authors' earlier free-perpetuity paper, but as a previously established mathematical object rather than a circularly defined one.

assumptions (9)

standard math Goldie implicit renewal theorem [15, Theorem 4.1]
Used in Theorem 5.1 proof to obtain the exact t^{-η} tail constant of X_11; its hypotheses are checked as conditions (iii-a)-(iii-c), (iv), (v).
standard math Vector perpetuity existence and moment bounds [8, Theorem 4.1.4, Remark 4.4.3]
Provides existence/uniqueness through vectorization and the p-th moment bounds used in Lemmas 3.1, 5.5, and 5.7.
standard math Free perpetuity existence and uniqueness from authors' prior paper [5, Theorems 4.6(ii), 3.15, 4.2]
Defines the limiting free perpetuity in Theorem 6.2. This is a self-citation, but the current paper proves convergence to that object rather than assuming the conclusion.
standard math Kuratowski–Ryll-Nardzewski measurable selection theorem
Used in Appendix C and Lemma 4.2/4.12 to construct measurable polar decompositions A=|A|U.
domain assumption Orthogonal invariance and positive semidefiniteness of the pair (A,B)
Structural assumption of Theorem 5.1; it makes the corner X_11 a scalar perpetuity and gives the principal-submatrix compression formulas via Corollary 4.5.
domain assumption Moment and non-arithmetic conditions (iii)-(v) of Theorem 5.1
Existence of η with E[A_11^η]=1 sets the tail index; E[det(A[2])^{η/2}]<1 ensures the second eigenvalue is subdominant; non-arithmeticity is required by Goldie's theorem.
domain assumption Asymptotic freeness condition (iii) in Theorem 6.2
Ensures finite joint moments of independent copies converge to free copies; the paper notes this is standard under orthogonal invariance, but it is explicitly assumed rather than proved.
standard math Known Wishart and GIG submatrix facts [23,24]
Used in Appendix B to derive the principal-submatrix law of the matrix Beta prime distribution.
standard math Coulomb gas large deviations [13]
Used in Appendix A to prove weak convergence of the Beta prime empirical spectral distribution to the limiting density.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the empirical spectral distribution of matrix perpetuities." pith.science (2026). https://pith.science/paper/66UVSMD6

@misc{pith2026260531054,
  author       = {Pith},
  title        = {Pith review of: On the empirical spectral distribution of matrix perpetuities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66UVSMD6}},
  note         = {Machine review of arXiv:2605.31054}
}

abstract

We study matrix perpetuities, that is, solutions to affine fixed-point equations of the form \[ \mathbf{X} \stackrel{d}{=} \mathbf{A}\,\mathbf{X} \,\mathbf{A}^\top+\mathbf{B},\qquad (\mathbf{A},\mathbf{B})\mbox{ and }\mathbf{X} \mbox{ are independent}, \] with particular emphasis on the empirical spectral distribution of the solution. We first establish existence and uniqueness results by relating the problem to classical vector perpetuities. Under orthogonal invariance, we then prove a compression identity for symmetric multiplicative convolution and show that principal submatrices of a matrix perpetuity are themselves lower-dimensional matrix perpetuities. For positive semidefinite, orthogonally invariant models, we prove a finite-dimensional spectral Kesten theorem: we obtain precise power-law tail asymptotics for the expected empirical spectral distribution, show that its tail is governed by the largest eigenvalue, and relate it explicitly to the tail of any diagonal entry. We also prove that, in the subcritical regime, the expected empirical spectral distribution of matrix perpetuities converges weakly, as the dimension tends to infinity, to the distribution of the corresponding free perpetuity. Our results are illustrated by matrix Beta prime perpetuities, for which explicit limiting spectral distributions are available.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Finite free perpetuities
math.PR 2026-06 unverdicted novelty 7.0 of 10

Finite free perpetuities are defined as degree-n monic polynomials solving a truncated perpetuity equation; the paper proves existence, uniqueness, real nonnegative zeros for admissible (A,B), and weak convergence of ...

Reference graph

Works this paper leans on

29 extracted references · 1 linked inside Pith · cited by 1 Pith paper

[5]

Belinschi, B

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

arXiv 2025
[1]

Anderson, Alice Guionnet, and Ofer Zeitouni.An introduction to random matrices, volume 118 ofCambridge Studies in Advanced Mathematics

Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni.An introduction to random matrices, volume 118 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010

2010
[2]

Matrix Whittaker processes.Probab

Jonas Arista, Elia Bisi, and Neil O’Connell. Matrix Whittaker processes.Probab. Theory Related Fields, 187(1-2):203–257, 2023

2023
[3]

Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices.Ann

Jonas Arista, Elia Bisi, and Neil O’Connell. Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices.Ann. Inst. Henri Poincar´ e Probab. Stat., 60(2):923–945, 2024

2024
[4]

Stationary inverse-Wishart polymers

Guillaume Barraquand and Zikun Ouyang. Stationary inverse-Wishart polymers. arXiv:2511.14375, 2025

arXiv 2025
[6]

Belinschi and Alexandru Nica

Serban T. Belinschi and Alexandru Nica. On a remarkable semigroup of homo- morphisms with respect to free multiplicative convolution.Indiana Univ. Math. J., 57(4):1679–1713, 2008

2008
[7]

Strict stationarity of generalized autoregressive processes.Ann

Philippe Bougerol and Nico Picard. Strict stationarity of generalized autoregressive processes.Ann. Probab., 20(4):1714–1730, 1992

1992
[8]

Buraczewski, E

D. Buraczewski, E. Damek, and T. Mikosch.Stochastic models with power-law tails. Springer Series in Operations Research and Financial Engineering. Springer, [Cham],

Show all 29 references

[9]

Convergence to stable laws for a class of multidimensional stochastic recursions.Probab

Dariusz Buraczewski, Ewa Damek, and Yves Guivarc’h. Convergence to stable laws for a class of multidimensional stochastic recursions.Probab. Theory Related Fields, 148(3-4):333–402, 2010

2010
[10]

Tail-homogeneity of stationary measures for some multidimensional stochastic recursions.Probab

Dariusz Buraczewski, Ewa Damek, Yves Guivarc’h, Andrzej Hulanicki, and Roman Urban. Tail-homogeneity of stationary measures for some multidimensional stochastic recursions.Probab. Theory Related Fields, 145(3-4):385–420, 2009

2009
[11]

Matsumoto-Yor processes on Jordan algebras

Reda Chhaibi and Manon Defosseux. Matsumoto-Yor processes on Jordan algebras. Probab. Theory Relat. Fields, 2025

2025
[12]

Damek and S

E. Damek and S. Mentemeier. Analysing heavy-tail properties of stochastic gradient descent by means of stochastic recurrence equations.J. Appl. Probab., page 1–25, 2026

2026
[13]

D. F´ eral. On large deviations for the spectral measure of discrete Coulomb gas. In S´ eminaire de probabilit´ es XLI, volume 1934 ofLecture Notes in Math., pages 19–49. Springer, Berlin, 2008

1934
[14]

Matrix Kesten recursion, inverse-Wishart ensemble and fermions in a Morse potential.J

Tristan Gauti´ e, Jean-Philippe Bouchaud, and Pierre Le Doussal. Matrix Kesten recursion, inverse-Wishart ensemble and fermions in a Morse potential.J. Phys. A, 54(25):Paper No. 255201, 58, 2021

2021
[15]

Charles M. Goldie. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., 1(1):126–166, 1991. 40 B. KO LODZIEJEK AND K. SZPOJANKOWSKI

1991
[16]

Guivarc’h and ´E

Y. Guivarc’h and ´E. Le Page. Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions.Ann. Inst. Henri Poincar´ e Probab. Stat., 52(2):503–574, 2016

2016
[17]

Hiai and D

F. Hiai and D. Petz.The semicircle law, free random variables and entropy, vol- ume 77 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000

2000
[18]

Liam Hodgkinson, Zhichao Wang, and Michael W. Mahoney. Models of heavy- tailed mechanistic universality. InForty-second International Conference on Machine Learning, 2025

2025
[19]

Horn and Charles R

Roger A. Horn and Charles R. Johnson.Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013

2013
[20]

In- verse problems for regular variation of linear filters, a cancellation property forσ-finite measures and identification of stable laws.Ann

Martin Jacobsen, Thomas Mikosch, Jan Rosi´ nski, and Gennady Samorodnitsky. In- verse problems for regular variation of linear filters, a cancellation property forσ-finite measures and identification of stable laws.Ann. Appl. Probab., 19(1):210–242, 2009

2009
[21]

H. Kesten. Random difference equations and renewal theory for products of random matrices.Acta Math., 131:207–248, 1973

1973
[22]

Ko lodziejek

B. Ko lodziejek. The Matsumoto-Yor property and its converse on symmetric cones. J. Theoret. Probab., 30(2):624–638, 2017

2017
[23]

An independence property for the product of GIG and gamma laws.Ann

G´ erard Letac and Jacek Weso lowski. An independence property for the product of GIG and gamma laws.Ann. Probab., 28(3):1371–1383, 2000

2000
[24]

The Matsumoto-Yor property and the struc- ture of the Wishart distribution.J

H´ el` ene Massam and Jacek Weso lowski. The Matsumoto-Yor property and the struc- ture of the Wishart distribution.J. Multivariate Anal., 97(1):103–123, 2006

2006
[25]

C. M. Newman. The distribution of Lyapunov exponents: exact results for random matrices.Comm. Math. Phys., 103(1):121–126, 1986

1986
[26]

On the multiplication of freeN-tuples of noncommutative random variables.Amer

Alexandru Nica and Roland Speicher. On the multiplication of freeN-tuples of noncommutative random variables.Amer. J. Math., 118(4):799–837, 1996

1996
[27]

E. B. Saff and V. Totik.Logarithmic potentials with external fields, volume 316 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Math- ematical Sciences]. Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom

1997
[28]

Fractional free convolution powers.Indiana Univ

Dimitri Shlyakhtenko and Terence Tao. Fractional free convolution powers.Indiana Univ. Math. J., 71(6):2551–2594, 2022

2022
[29]

H. Yoshida. Remarks on a free analogue of the beta prime distribution.J. Theoret. Probab., 33(3):1363–1400, 2020. Bartosz Ko lodziejek: F aculty of Mathematics and Information Sciences, W arsaw Uni- versity of Technology, Koszykowa 75, 00-662 W arsaw, Poland Email address:bart...

2020

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [5]

Belinschi, B

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

arXiv 2025

[2] [1]

Anderson, Alice Guionnet, and Ofer Zeitouni.An introduction to random matrices, volume 118 ofCambridge Studies in Advanced Mathematics

Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni.An introduction to random matrices, volume 118 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010

2010

[3] [2]

Matrix Whittaker processes.Probab

Jonas Arista, Elia Bisi, and Neil O’Connell. Matrix Whittaker processes.Probab. Theory Related Fields, 187(1-2):203–257, 2023

2023

[4] [3]

Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices.Ann

Jonas Arista, Elia Bisi, and Neil O’Connell. Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices.Ann. Inst. Henri Poincar´ e Probab. Stat., 60(2):923–945, 2024

2024

[5] [4]

Stationary inverse-Wishart polymers

Guillaume Barraquand and Zikun Ouyang. Stationary inverse-Wishart polymers. arXiv:2511.14375, 2025

arXiv 2025

[6] [6]

Belinschi and Alexandru Nica

Serban T. Belinschi and Alexandru Nica. On a remarkable semigroup of homo- morphisms with respect to free multiplicative convolution.Indiana Univ. Math. J., 57(4):1679–1713, 2008

2008

[7] [7]

Strict stationarity of generalized autoregressive processes.Ann

Philippe Bougerol and Nico Picard. Strict stationarity of generalized autoregressive processes.Ann. Probab., 20(4):1714–1730, 1992

1992

[8] [8]

Buraczewski, E

D. Buraczewski, E. Damek, and T. Mikosch.Stochastic models with power-law tails. Springer Series in Operations Research and Financial Engineering. Springer, [Cham],