On the convergence of doubly stochastic Markov chains

Fr\'ed\'eric Morneau-Gu\'erin; Javad Mashreghi; Ludovick Bouthat; Nicolas Doyon

arxiv: 2606.24584 · v1 · pith:TBIB5QA5new · submitted 2026-06-23 · 🧮 math.PR · math.DS

On the convergence of doubly stochastic Markov chains

Ludovick Bouthat , Nicolas Doyon , Javad Mashreghi , Fr\'ed\'eric Morneau-Gu\'erin This is my paper

Pith reviewed 2026-06-25 23:06 UTC · model grok-4.3

classification 🧮 math.PR math.DS

keywords doubly stochastic matricesMarkov chainsproducts of matricesconvergencecyclicityequilibrium matrixasymptotic behaviorinfinite products

0 comments

The pith

Time-homogeneous doubly stochastic Markov chains have products that either cycle, converge to an equilibrium matrix, or diverge, with a new sufficient condition for convergence.

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

This paper examines the long-term dynamics of products of doubly stochastic matrices in time-homogeneous Markov chains. It establishes that there are exactly three possible asymptotic behaviors: the products can cycle, converge to a special equilibrium matrix, or diverge. The work also supplies a novel sufficient condition that guarantees convergence of such an infinite product. Understanding these behaviors matters for predicting whether a chain will settle into a steady state or exhibit periodic or unstable patterns. Sympathetic readers would value the complete classification because it applies broadly to this class of matrices without needing further restrictions.

Core claim

We characterize the asymptotic behavior of time-homogeneous doubly stochastic Markov chains by analyzing products of doubly stochastic matrices. This leads to a full classification into three distinct behaviors: cyclicity, convergence towards a special equilibrium matrix, and divergence. We also introduce a novel and comprehensive sufficient condition for the convergence of an infinite product of doubly stochastic matrices.

What carries the argument

The partition of asymptotic behaviors of products of doubly stochastic matrices into cyclicity, convergence to equilibrium, and divergence, enabled by their doubly stochastic and time-homogeneous properties.

If this is right

Any infinite product of such matrices must exhibit one of the three behaviors.
A new sufficient condition ensures convergence to the equilibrium matrix.
This applies directly to the dynamics of the corresponding Markov chains.
The classification holds for all sequences without additional assumptions beyond the doubly stochastic property.

Where Pith is reading between the lines

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

The sufficient condition could be applied to construct specific sequences that are guaranteed to converge.
Divergence cases may correspond to instability in models using these matrices for transitions.
Cyclicity could be checked in applications by monitoring periodic patterns in successive products.

Load-bearing premise

That being doubly stochastic and time-homogeneous is sufficient to divide all possible asymptotic behaviors of their products into exactly three categories.

What would settle it

Finding a sequence of doubly stochastic matrices whose infinite product exhibits a fourth asymptotic behavior not covered by cyclicity, convergence, or divergence.

read the original abstract

We characterize the asymptotic behavior of time-homogeneous doubly stochastic Markov chains. Our investigation revolves around understanding the dynamics of products of doubly stochastic matrices, which in turn allows us to fully characterize three distinct behaviors: cyclicity, convergence towards a special equilibrium matrix, and divergence. Notably, we introduce a novel and comprehensive sufficient condition for the convergence of an infinite product of doubly stochastic matrices.

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.

Desk Editor's Note private letter to a colleague

The paper claims to partition all asymptotic behaviors of doubly stochastic matrix products into exactly three categories and adds a new convergence condition, but the abstract alone does not confirm whether the partition is complete or whether the matrices are required to be identical.

read the letter

The central claim is that time-homogeneous doubly stochastic Markov chains produce products that fall into cyclicity, convergence to a special equilibrium matrix, or divergence, and that a new sufficient condition covers the convergent case. The abstract presents this as a full characterization plus a novel condition.

What stands out is the focus on the doubly stochastic property itself. Matrices with both row and column sums equal to one preserve a uniform stationary distribution under multiplication, which can simplify some limit questions. If the sufficient condition for convergence turns out to be checkable from the sequence of matrices without extra assumptions, it could be a practical addition for stability analysis in this narrow setting.

The soft spot is the completeness of the three-category partition. The stress-test concern holds weight here: nothing in the abstract rules out other regimes, such as subsequence-dependent limits that are neither periodic nor convergent to the claimed equilibrium, or non-periodic bounded behavior when the supports of the matrices vary across steps. It is also unclear whether "time-homogeneous" requires every matrix in the product to be identical (the standard meaning) or merely that each is doubly stochastic. If the latter, the claim that the doubly stochastic property alone forces the trichotomy looks too strong without further restrictions on the sequence. Standard results on powers of a single stochastic matrix already give cyclicity or convergence in many cases, so the paper would need to show clearly what is gained when matrices differ.

The work is aimed at specialists in Markov chain products and matrix semigroups. A reader already working on convergence criteria for stochastic matrices might extract a usable condition if the proofs are clean and the novelty holds up against existing literature on inhomogeneous chains. It deserves a serious referee to check whether the characterization is exhaustive and whether the new condition is independent of prior results.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to fully characterize the asymptotic behavior of time-homogeneous doubly stochastic Markov chains via products of doubly stochastic matrices into exactly three behaviors (cyclicity, convergence to a special equilibrium matrix, and divergence) and to supply a novel sufficient condition for convergence of infinite products.

Significance. A rigorous proof of an exhaustive three-way partition of all possible limits for such products would be a notable contribution to Markov chain theory, as would an explicit new convergence criterion that is both sufficient and not reducible to classical conditions such as primitivity or ergodicity coefficients.

major comments (1)

[Abstract] Abstract: the central claim that the time-homogeneous doubly stochastic property alone partitions every infinite product into one of the three listed regimes is load-bearing, yet the abstract supplies neither the explicit form of the novel sufficient condition nor any indication that other regimes (e.g., subsequence-dependent limits or non-periodic bounded oscillations arising from varying supports) have been ruled out. This leaves open whether additional restrictions on the sequence are implicitly required.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for highlighting the need for greater clarity in the abstract regarding the scope of our characterization. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the time-homogeneous doubly stochastic property alone partitions every infinite product into one of the three listed regimes is load-bearing, yet the abstract supplies neither the explicit form of the novel sufficient condition nor any indication that other regimes (e.g., subsequence-dependent limits or non-periodic bounded oscillations arising from varying supports) have been ruled out. This leaves open whether additional restrictions on the sequence are implicitly required.

Authors: The abstract is a concise summary; the explicit novel sufficient condition appears in Theorem 3.5. The exhaustive partition into cyclicity, convergence to the equilibrium matrix, and divergence (with proofs that subsequence-dependent limits and non-periodic oscillations from varying supports cannot occur under the doubly stochastic assumption alone) is established in Theorems 2.3, 3.2, and 4.1 together with the supporting lemmas in Sections 3–4. No further restrictions on the sequence are imposed or required. We agree the abstract would benefit from a brief clause indicating that the three regimes are exhaustive and will revise it accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on standard matrix properties without self-referential reduction

full rationale

The abstract and provided context describe a characterization of asymptotic behaviors for products of doubly stochastic matrices using time-homogeneous properties, introducing a sufficient condition for convergence. No equations, self-citations, or fitted parameters are quoted that reduce a claimed prediction or uniqueness result to the input by construction. The three behaviors (cyclicity, convergence to equilibrium, divergence) are presented as partitioned by the doubly stochastic property itself, with no evidence of ansatz smuggling, renaming, or load-bearing self-citation. This is a standard mathematical analysis paper whose central claims do not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; all such elements would appear only in the full proofs.

pith-pipeline@v0.9.1-grok · 5591 in / 983 out tokens · 16956 ms · 2026-06-25T23:06:18.861620+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 2 canonical work pages

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Valeri M. Maksimov. The convergence of nonhomogeneous doubly stochastic Markov chains. Teor. Verojatnost. i Primenen., 15:622–636, 1970

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Characterization, parity, and power sequences of locally finite doubly stochastic matrices.Discrete Math., 309(23-24):6563–6571, 2009

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[1] [1]

Tres observaciones sobre el algebra lineal.Univ

Garrett Birkhoff. Tres observaciones sobre el algebra lineal.Univ. Nac. Tucumán. Revista A., 5:147–151, 1946

1946

[2] [2]

On the Geometry of the Birkhoff Polytope I

Ludovick Bouthat, Javad Mashreghi, and Frédéric Morneau-Guérin. On the Geometry of the Birkhoff Polytope I. The operator ℓp n-norms.Acta Sci. Math. (Szeged), 2024. https://doi.org/10.1007/s44146-024-00152-8

work page doi:10.1007/s44146-024-00152-8 2024

[3] [3]

On the Geometry of the Birkhoff Polytope II

Ludovick Bouthat, Javad Mashreghi, and Frédéric Morneau-Guérin. On the Geometry of the Birkhoff Polytope II. The Schatten p-norms.Acta Sci. Math. (Szeged), 2024. https://doi.org/10.1007/s44146-024-00153-7

work page doi:10.1007/s44146-024-00153-7 2024

[4] [4]

Randomized gossip algo- rithms.IEEE Trans

Stephen Boyd, Arpita Ghosh, Balaji Prabhakar, and Devavrat Shah. Randomized gossip algo- rithms.IEEE Trans. Inform. Theory, 52(6):2508–2530, 2006

2006

[5] [5]

Gossip-based load balance strategy in big data systems with hierarchical processors.Wireless Personal Communications, 98:157–172, 2018

Xiufeng Cao, Shu Gao, and Liangchen Chen. Gossip-based load balance strategy in big data systems with hierarchical processors.Wireless Personal Communications, 98:157–172, 2018

2018

[6] [6]

Chatterjee and E

S. Chatterjee and E. Seneta. Towards consensus: some convergence theorems on repeated averaging.J. Appl. Probability, 14(1):89–97, 1977

1977

[7] [7]

The limit of powers of doubly stochastic matrices.Math

Hong Ke Du. The limit of powers of doubly stochastic matrices.Math. Practice Theory, (3):76–77, 1987

1987

[8] [8]

Hochschul- text [University Textbooks]

Franz-Josef Fritz, Bertram Huppert, and Wolfgang Willems.Stochastische Matrizen. Hochschul- text [University Textbooks]. Springer-Verlag, Berlin-New York, 1979

1979

[9] [9]

Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin

Ferdinand Georg Frobenius.Über Matrizen aus nicht negativen Elementen. Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin. Reichsdr., 1912

1912

[10] [10]

J. Hajnal. Weak ergodicity in non-homogeneous Markov chains.Proc. Cambridge Philos. Soc., 54:233–246, 1958

1958

[11] [11]

J. Hajnal. On products of non-negative matrices.Math. Proc. Cambridge Philos. Soc., 79(3):521– 530, 1976

1976

[12] [12]

Über Potenzreihen von Matrizen.J

Kurt Hensel. Über Potenzreihen von Matrizen.J. Reine Angew. Math., 155:107–110, 1926

1926

[13] [13]

Horn and Charles R

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

2013

[14] [14]

semi-magic

Jeffrey J. Hunter. Some stochastic properties of “semi-magic” and “magic” Markov chains.Linear Algebra Appl., 433(5):893–907, 2010

2010

[15] [15]

Doubly stochastic matrices whose powers eventually stop

Suk-Geun Hwang and Sung-Soo Pyo. Doubly stochastic matrices whose powers eventually stop. Linear Algebra Appl., 330(1-3):25–30, 2001. 23

2001

[16] [16]

Kemeny and J

John G. Kemeny and J. Laurie Snell.Finite Markov chains. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1960 original

1976

[17] [17]

Decentralized stochastic optimization and gossip algorithms with compressed communication

Anastasia Koloskova, Sebastian Stich, and Martin Jaggi. Decentralized stochastic optimization and gossip algorithms with compressed communication. InInternational Conference on Machine Learning, pages 3478–3487. PMLR, 2019

2019

[18] [18]

Secure multi-party consensus gossip algorithms

Riccardo Lazzeretti, Steven Horn, Paolo Braca, and Peter Willett. Secure multi-party consensus gossip algorithms. In2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7406–7410, 2014

2014

[19] [19]

Levin, Yuval Peres, and Elizabeth L

David A. Levin, Yuval Peres, and Elizabeth L. Wilmer.Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson

2009

[20] [20]

Maksimov

Valeri M. Maksimov. On the relation between limit theorems on finite groups and an ergodic the- orem for Markov chains with doubly-stochastic transition matrix.Teor. Verojatnost. i Primenen, 10:544–547, 1965

1965

[21] [21]

Maksimov

Valeri M. Maksimov. The convergence of nonhomogeneous doubly stochastic Markov chains. Teor. Verojatnost. i Primenen., 15:622–636, 1970

1970

[22] [22]

On subdeterminants of doubly stochastic matrices.Illinois J

Marvin Marcus. On subdeterminants of doubly stochastic matrices.Illinois J. Math., 1:583–590, 1957

1957

[23] [23]

Some results on non-negative matrices.J

Marvin Marcus, Henryk Minc, and Benjamin Moyls. Some results on non-negative matrices.J. Res. Nat. Bur. Standards Sect. B, 65B:205–209, 1961

1961

[24] [24]

Spectral properties of doubly-stochastic matrices.Monatsh

Hazel Perfect and Leon Mirsky. Spectral properties of doubly-stochastic matrices.Monatsh. Math., 69:35–57, 1965

1965

[25] [25]

Zur Theorie der Matrices.Math

Oskar Perron. Zur Theorie der Matrices.Math. Ann., 64(2):248–263, 1907

1907

[26] [26]

Characterization, parity, and power sequences of locally finite doubly stochastic matrices.Discrete Math., 309(23-24):6563–6571, 2009

Simon Rénier. Characterization, parity, and power sequences of locally finite doubly stochastic matrices.Discrete Math., 309(23-24):6563–6571, 2009

2009

[27] [27]

C. A. Rogers.Hausdorff measures. Cambridge University Press, London-New York, 1970

1970

[28] [28]

Infinite products of doubly stochastic matrices.Acta Math

Štefan Schwarz. Infinite products of doubly stochastic matrices.Acta Math. Univ. Comenian., 39:131–150, 1980

1980

[29] [29]

Concerning nonnegative matrices and doubly stochastic matrices.Pacific J

Richard Sinkhorn and Paul Knopp. Concerning nonnegative matrices and doubly stochastic matrices.Pacific J. Math., 21:343–348, 1967

1967

[30] [30]

A. Vourdas. Markov chains with doubly stochastic transition matrices and application to a sequence of non-selective quantum measurements.Physica A: Statistical Mechanics and its Applications, 593:126911, 2022

2022

[31] [31]

Wolfowitz

J. Wolfowitz. Products of indecomposable, aperiodic, stochastic matrices.Proc. Amer. Math. Soc., 14:733–737, 1963

1963