Functional dependence and synchronous coupling in ergodic autoregressions

Lionel Truquet; Paul Doukhan

Positive Lyapunov exponents can stop natural innovation couplings from forgetting initial conditions, even when the autoregressive chain is uniformly ergodic.

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2026-07-10 05:13 UTC pith:WEOMLWDB

load-bearing objection Clean short note showing that natural-innovation functional dependence can fail for uniformly ergodic AR maps precisely when the Lyapunov exponent turns positive; the math holds and the warning is useful.

arxiv 2607.08567 v1 pith:WEOMLWDB submitted 2026-07-09 math.ST stat.TH

Functional dependence and synchronous coupling in ergodic autoregressions

Paul Doukhan , Lionel Truquet This is my paper

classification math.ST stat.TH MSC 60J0537H1562M10

keywords functional dependenceBernoulli shift representationiterated random mapsLyapunov exponentsynchronous couplinguniform ergodicityautoregressive processes

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Functional dependence coefficients for nonlinear time series are defined from a Bernoulli-shift representation driven by innovations, so they are properties of a chosen probability space rather than of the process law alone. This note constructs uniformly ergodic autoregressive models for which the natural synchronous coupling—two trajectories driven by identical innovations after a single initial perturbation—fails to forget that perturbation whenever an associated Lyapunov exponent is positive. The transition in coupling behavior tracks the sign change of that exponent. The authors show that a causal Bernoulli representation with geometrically decaying coefficients always exists after the probability space is enlarged by Doeblin regeneration, but the natural innovation space itself need not work. The practical consequence is that limit theorems relying on summable functional dependence must be checked against the representation actually used.

Core claim

For the smooth AR(1) X_t = r(X_{t-1}) + ε_t with bounded C¹ derivative, a positive Lyapunov exponent λ = E log |r'(X_0)| > 0 together with mild regularity (including a local control on the negative log of the expansion ratio Q_t) implies that the natural synchronous coupling distance D_n does not tend to zero almost surely. Consequently the natural-innovation functional dependence coefficients cannot be summable, even though the Markov chain is uniformly ergodic and φ-mixing at a geometric rate. The same transition appears for the explicit family X_t = a f(X_{t-1}) + ε_t with smooth periodic f once |a| is large enough that λ_a > 0.

What carries the argument

The natural (forward) synchronous coupling D_t of two trajectories sharing the same innovation sequence after one initial perturbation, together with the Lyapunov exponent λ = ∫ E log |f'_1(y)| π(dy). Theorem 3.1 converts positivity of λ plus a uniform-continuity and integrability condition into the almost-sure failure of |D_n| o 0, which immediately rules out summable functional dependence on the natural innovation space.

Load-bearing premise

The technical bound that the expected negative logarithm of the local expansion ratio vanishes uniformly whenever the derivative is small and the two trajectories are already close; if that control fails, the argument that a positive Lyapunov exponent blocks synchronization collapses.

What would settle it

Simulate the natural synchronous coupling for X_t = a sin(X_{t-1}) + standard Gaussian noise at a value of |a| (for example near 6) where the empirical Lyapunov exponent is clearly positive; if the distance |D_n| tends to zero with positive probability, the central claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Natural-innovation functional dependence coefficients need not decay for uniformly ergodic AR models once the Lyapunov exponent is positive.
A causal Bernoulli-shift representation with geometrically decaying coefficients always exists after the probability space is enlarged by Doeblin regeneration.
Pullback and forward synchronization of the natural coupling can both fail even when the chain is geometrically φ-mixing.
For the family X_t = a f(X_{t-1}) + ε_t with smooth periodic f, the sign of λ_a controls whether the natural coupling forgets initial perturbations.
Any statistical argument that invokes summable δ_q(n) must verify that the chosen innovation representation actually contracts.

Where Pith is reading between the lines

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

Negative Lyapunov exponents may still be insufficient for a natural Bernoulli representation outside global contractivity; the authors leave this open and it is a natural next test.
Threshold and other Foster–Lyapunov ergodic autoregressions are likely to exhibit the same representation gap, so natural-innovation dependence diagnostics should be treated cautiously there as well.
Empirical estimates of functional dependence obtained by driving two simulated paths with common noise should be accompanied by a check of the empirical Lyapunov exponent before decay rates are trusted.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Clean short note showing that natural-innovation functional dependence can fail for uniformly ergodic AR maps precisely when the Lyapunov exponent turns positive; the math holds and the warning is useful.

read the letter

The one thing worth knowing is that Doukhan and Truquet isolate a concrete, representation-dependent failure mode of Wu’s functional dependence: for a family of uniformly ergodic AR(1) maps X_t = a f(X_{t-1}) + ε_t with smooth periodic f, the natural synchronous coupling forgets initial conditions when the Lyapunov exponent is negative and fails to forget (almost surely) once it becomes positive. Theorem 3.1 plus Proposition 3.3 make that transition explicit; the discontinuous examples in Section 2 already show the same phenomenon without any Lyapunov machinery. That separation between the natural innovation space and an enlarged Doeblin space is the actual contribution, and it is cleanly done.

What works well is the transparency. Proposition 2.1 is elementary but useful: a causal Bernoulli representation on the natural innovations forces pullback synchronization, which the discontinuous maps immediately violate. The smooth case then uses only the ergodic theorem, absolute continuity under C^{1} maps, and a product representation of the difference D_n. Condition (5) is the only technical hypothesis; they verify it carefully for sine and cosine under a mild periodized density bound, and they do not pretend it holds for arbitrary r. Section 4 correctly recalls that every Doeblin chain admits a geometrically decaying representation after splitting, so the paper never claims non-existence of some Bernoulli shift—only that the natural one can be useless for dependence coefficients. Citations are standard and not load-bearing for the new theorems.

Soft spots are minor and proportional. The discontinuous examples are elementary (as the authors essentially admit). The threshold |a| > k_0 is only an upper bound on the transition; they note numerically that it may occur earlier. Condition (5) remains a standing assumption for general r, but that is honestly flagged and does not undermine the sine/cosine family or the discontinuous cases. No circularity, no free parameters, no over-claim.

This is for people who actually use functional dependence or synchronous couplings on nonlinear Markov chains. It will not reorganize the field, but it will stop a recurring misuse. I would send it to a serious referee without hesitation; the note is short, the proofs are self-contained, and the practical warning is solid. Engage with it if you work in weak dependence or random dynamical systems; otherwise it is optional reading.

Referee Report

0 major / 5 minor

Summary. The paper observes that functional dependence coefficients (in the sense of Wu) are representation-dependent and need not reflect the intrinsic ergodicity of a Markov chain when constructed from the natural innovation sequence. After two discontinuous AR examples where the natural synchronous coupling never forgets initial conditions, the authors treat the smooth model X_t = r(X_{t-1}) + ε_t with bounded C^1 r. Under a positive Lyapunov exponent λ = E log |r'(X_0)| > 0 together with uniform continuity of r', integrability of log^- |r'|, and a local control (5) on log^- Q_t, Theorem 3.1 proves that the forward coupling satisfies P(lim D_n = 0) = 0 (and likewise for Lebesgue-almost every pair of initial conditions). Consequently the natural-innovation functional dependence coefficients cannot be summable. An explicit periodic family r(x) = a f(x) is shown to have λ_a > 0 for large |a| while remaining uniformly ergodic for every a; Section 4 recalls that a geometrically decaying Bernoulli representation always exists after enlarging the probability space via Doeblin splitting.

Significance. The note cleanly isolates a representation-versus-distribution subtlety that is easy to overlook when applying functional dependence to iterated random maps. The proofs of Proposition 2.1, Theorem 3.1 and Proposition 3.3 are self-contained, rely only on the ergodic theorem, absolute continuity under C^1 maps, and elementary Cesàro estimates, and correctly link the sign of the Lyapunov exponent to the failure of natural synchronous coupling. The concrete sine/cosine verification of condition (5) and the numerical threshold k_0 ≈ 5.827 supply a usable example. The distinction drawn in Section 4 between the natural innovation space and an enlarged Doeblin space is pedagogically useful for the time-series community. These contributions are solid and of genuine interest for both probability and nonlinear time-series analysis.

minor comments (5)

Introduction, line after the display of δ_q(n): “geometric ergdicity” is a typo for “geometric ergodicity”.
Section 3.2, display (5) and the sentence that follows: the standing hypothesis is verified only for sine/cosine; a short remark on the class of r for which (5) is automatic (e.g., analytic periodic maps) would improve readability.
Section 3.3, Remark 3.6: the numerical value k_0 ≈ 5.827 is given without stating the quadrature method or precision; a one-line description would make the claim reproducible.
Throughout: occasional French spellings (“connexions”, “autor´egressifs”) and the future date “July 10, 2026” on the title page should be standardized for an English-language journal.
References: several entries lack DOIs or final page ranges; a quick consistency pass would help.

Circularity Check

0 steps flagged

No significant circularity: Theorem 3.1 and the Lyapunov transition are derived from first principles without self-definitional or fitted reductions.

full rationale

The paper's central claims (Proposition 2.1 obstruction via pullback synchronization; Theorem 3.1 that positive Lyapunov exponent λ = E log |r'(X0)| > 0 plus mild regularity implies P(lim Dn = 0) = 0 for the natural forward coupling; and the explicit AR(1) family of Proposition 3.3) are proved directly. The proof of Theorem 3.1 proceeds by contradiction from the product representation |Dn+1| = |D1| ∏ Qt, Cesàro control of log Qt - log |r'(Xt)| on the event {Dt o 0} via uniform continuity of r' and the local integrability hypothesis (5), and the ergodic theorem applied to the stationary chain; none of these steps is definitional of the conclusion or obtained by fitting. The discontinuous counter-examples of Section 2 already establish the representation-dependent failure mode without any Lyapunov machinery. Self-citations ([5], [7], [8], [3], [6]) are to earlier surveys, simulation studies or general weak-dependence monographs by the same authors and are not invoked as load-bearing uniqueness theorems or ansätze that force the new results. Section 4's Doeblin/Nummelin construction is classical and independent. No fitted parameters are renamed as predictions. The derivation is therefore self-contained against its own stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper rests on classical ergodic theory and Markov-chain minorization; no free parameters are fitted to data and no new physical or mathematical entities are postulated. The only non-standard technical hypothesis is the local integrability control (5), which is verified for the concrete examples.

axioms (4)

standard math Ergodic theorem for the stationary Markov chain yields the almost-sure limit of (1/n) sum log |f'_t(X_{t-1})| equal to the Lyapunov exponent λ.
Invoked in Section 3.1 to define λ and again in the proof of Theorem 3.1.
domain assumption Doeblin minorization (one-step) implies uniform ergodicity and geometric φ-mixing for the Markov chain.
Used throughout Sections 2–4; standard reference Meyn–Tweedie / Kulik.
ad hoc to paper Condition (5): lim_η↓0 sup_t E[log^- Q_t 1_{|r'|≤η,|D_t|≤δ_0}] = 0.
Standing technical hypothesis of Theorem 3.1; verified for sine/cosine via absolute continuity of Lebesgue integral (Remark 3.4).
domain assumption Noise density positive and bounded away from zero on compact sets (or the periodized bound (9)).
Guarantees absolute continuity of the invariant measure and the Doeblin condition for the AR models.

pith-pipeline@v1.1.0-grok45 · 15727 in / 2406 out tokens · 66890 ms · 2026-07-10T05:13:22.857715+00:00 · methodology

0 comments

read the original abstract

Functional dependence measures have become an important tool in the analysis of nonlinear time series and are typically formulated with respect to a given innovation representation of the process. This note points out that the probability space on which such representations yield the expected memory loss properties may not always coincide with the natural dynamical probability space of the model. We exhibit classes of uniformly ergodic autoregressive processes for which the behavior of the natural innovation coupling undergoes a qualitative transition as the model parameter varies. For this family of models, this transition coincides with a change in the sign of an associated Lyapunov exponent. In particular, a positive Lyapunov exponent may prevent the forgetting of initial perturbations along trajectories driven by the same innovations, despite uniform ergodicity of the associated Markov chain. These observations highlight the importance of carefully specifying the underlying probability space when interpreting or applying functional dependence measures.

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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work page 1984
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P. Nummelin and R. L. Tweedie. Geometric ergodicity and r-positivity for general markov chains.Ann. Prob., 6:404—-420, 1978

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[1] [1]

D. W. K. Andrews. Non strong mixing autoregressive processes.Journal of Applied Probability, 21:930–934, 1984

work page 1984

[2] [2]

Springer Monographs in Mathematics

Ludwig Arnold.Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg, 1998. ISBN 978-3-540-63758-1. doi: 10.1007/978-3-662-12878-7

work page doi:10.1007/978-3-662-12878-7 1998

[3] [3]

Dedecker, P

J. Dedecker, P. Doukhan, G. Lang, J-R. Leon, S. Louhichi, and C. Prieur.Weak dependence: with examples and applications. Lecture notes in statistics. Springer, New York, 2007. ISBN 978-0-387-69951-6

work page 2007

[4] [4]

Diaconis and D

P. Diaconis and D. Freedman. Iterated random functions.SIAM review, 41(1):45–76, 1999

work page 1999

[5] [5]

P. Doukhan. Simulation in the general first order autoregressive process (unidimensional normal case). InSpecifying Statistical Models: From Parametric to Non-Parametric, Using Bayesian or Non-Bayesian Approaches, pages 50–68. 1983

work page 1983

[6] [6]

Doukhan.Mixing, properties and models, volume 85 ofLecture Notes in Statistics

P. Doukhan.Mixing, properties and models, volume 85 ofLecture Notes in Statistics. Springer- Verlag, New York, 1994

work page 1994

[7] [7]

Doukhan and M

P. Doukhan and M. Ghindes. Processus autor´ egressifs non-lin´ eaires.Comptes Rendus de l’Acad´ emie des Sciences. Paris, S´ erie A, 290:921–923, 1980. 12

work page 1980

[8] [8]

Doukhan, N

P. Doukhan, N. Mayo, and L. Truquet. Weak dependence, models and some applications. Metrika, 69(2):199–225, 2009

work page 2009

[9] [9]

Flandoli, B

F. Flandoli, B. Gess, and M. Scheutzow. Synchronization by noise for order-preserving random dynamical systems.The Annals of Probability, pages 1325–1350, 2017

work page 2017

[10] [10]

Synchronization by noise.Proba- bility Theory and Related Fields, 168(3):511–556, 2017

Franco Flandoli, Benjamin Gess, and Michael Scheutzow. Synchronization by noise.Proba- bility Theory and Related Fields, 168(3):511–556, 2017

work page 2017

[11] [11]

Lyapunov exponents and synchronisation by noise for systems of spdes.The Annals of Probability, 52(5):1903–1953, 2024

Benjamin Gess and Pavlos Tsatsoulis. Lyapunov exponents and synchronisation by noise for systems of spdes.The Annals of Probability, 52(5):1903–1953, 2024

work page 1903

[12] [12]

De Gruyter Studies in Mathematics, 2018

Alexei Kulik.Ergodic Behavior of Markov Processes With Applications to Limit Theorems, volume 67. De Gruyter Studies in Mathematics, 2018

work page 2018

[13] [13]

G. Letac. A contraction principle for certain Markov chains and its applications. In H. Kesten et C. M. Newman J. E. Cohen, editor,Random matrices and their applications, volume 50, pages 263–273. 1984

work page 1984

[14] [14]

Meyn and Richard L

Sean P. Meyn and Richard L. Tweedie.Markov Chains and Stochastic Stability. Communi- cations and Control Engineering. Springer, London, 1993. ISBN 9783540198642

work page 1993

[15] [15]

Mokkadem

A. Mokkadem. Sur un mod` ele autor´ egressif non lin´ eaire, ergodicit´ e geometrique et mixing. Journal of Time Series Analysis, 2:195–204, 1987

work page 1987

[16] [16]

Necessary and sufficient conditions for stable synchronization in random dynamical systems.Ergodic Theory and Dynamical Systems, 38(5):1857–1875, 2018

Julian Newman. Necessary and sufficient conditions for stable synchronization in random dynamical systems.Ergodic Theory and Dynamical Systems, 38(5):1857–1875, 2018

work page 2018

[17] [17]

Nummelin and R

P. Nummelin and R. L. Tweedie. Geometric ergodicity and r-positivity for general markov chains.Ann. Prob., 6:404—-420, 1978

work page 1978

[18] [18]

D. Ruelle. Applications conservant une mesure absolument continue par rapport ` adxsur [0,1].Communications in Mathematical Physics, 55:47–51, 1977

work page 1977

[19] [19]

Synchronization, lyapunov exponents and stable manifolds for random dynamical systems

Michael Scheutzow and Isabell Vorkastner. Synchronization, lyapunov exponents and stable manifolds for random dynamical systems. InInternational Conference on Stochastic Partial Differential Equations and Related Fields, pages 359–366. Springer, 2016

work page 2016

[20] [20]

W. B. Wu. Nonlinear system theory: Another look at dependence.Proceedings of the National Academy of Science, 102(40):14150–14154, 2005

work page 2005

[21] [21]

Limit theorems for iterated random functions.Journal of Applied Probability, 41(2):425–436, 2004

Wei Biao Wu and Xiaofeng Shao. Limit theorems for iterated random functions.Journal of Applied Probability, 41(2):425–436, 2004. 13

work page 2004