pith. sign in

arxiv: 2604.17203 · v1 · submitted 2026-04-19 · 🧮 math.DS · math.PR

Functional correlation bound for random Lasota--Yorke maps with holes and its applications to conditional normal approximations

Juho Lepp\"anen , Yuto Nakajima , Yushi Nakano This is my paper

Pith reviewed 2026-05-10 06:21 UTC · model grok-4.3

classification 🧮 math.DS math.PR
keywords functional correlation boundsLasota-Yorke mapsopen dynamical systemsrandom maps with holesconditional central limit theoremWasserstein distanceKolmogorov distanceexponential decay
0
0 comments X

The pith

A functional correlation bound with exponential decay holds for random Lasota-Yorke maps with holes, yielding conditional central limit theorems with explicit rates.

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

The paper establishes that functional correlation bounds can be extended from closed dynamical systems to random open ones generated by families of Lasota-Yorke maps with holes. This requires new Lasota-Yorke type inequalities to handle the loss of mass from escaping trajectories. With this bound in hand, the authors derive a conditional central limit theorem with rates in Wasserstein distance, a conditional functional central limit theorem in an integral distance over smooth test functions, and a Kolmogorov distance bound obtained via Tikhomirov's method. If correct, these results provide quantitative control on the distribution of sums along surviving orbits in transient chaotic systems.

Core claim

We show that the framework of functional correlation bounds, originally developed for closed systems, can be adapted to this random open setting by introducing new ingredients based on Lasota-Yorke type inequalities in order to control the effect of escaping trajectories. We establish an FCB with exponential decay and combine it with abstract normal-approximation results to obtain a conditional CLT with rates in Wasserstein distance and a conditional functional CLT with a rate in an integral distance over Barbour's class of smooth test functions. Additionally, we adapt Tikhomirov's method to obtain a bound in Kolmogorov distance for the conditional CLT.

What carries the argument

The functional correlation bound (FCB) with exponential decay, obtained by adapting the closed-system version through Lasota-Yorke inequalities that bound the contribution of escaping trajectories.

If this is right

  • Normalized sums of observables along surviving orbits converge in distribution to a normal law at an explicit Wasserstein rate.
  • The process of partial sums satisfies a conditional functional CLT with a rate measured in an integral distance over a class of smooth test functions.
  • A Kolmogorov distance bound for the conditional CLT follows from an adaptation of Tikhomirov's method.

Where Pith is reading between the lines

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

  • The same adaptation technique may apply to other piecewise expanding maps if analogous inequalities can be established.
  • The explicit rates could be used to quantify the accuracy of Monte Carlo estimates for escape rates or conditional expectations in open systems.
  • Extensions to non-exponential decay rates or to systems with holes of varying size would test the robustness of the approach.

Load-bearing premise

Lasota-Yorke type inequalities suffice to control the loss of mass from escaping trajectories in the random open system.

What would settle it

A specific family of random Lasota-Yorke maps with holes and a pair of test functions for which the correlation does not decay exponentially would falsify the bound.

read the original abstract

This paper investigates the statistical properties of random open dynamical systems generated by families of Lasota--Yorke maps. Open systems, in which trajectories may escape through `holes', model transient phenomena and present additional difficulties for statistical analysis because the underlying ensemble loses mass over time. We show that the framework of functional correlation bounds (FCB), originally developed for closed systems, can also be adapted to this random open setting. The extension requires new ingredients based on Lasota--Yorke type inequalities in order to control the effect of escaping trajectories. We establish an FCB with exponential decay and combine it with the abstract normal-approximation results of \cite{LNN25,LS20} to obtain a conditional CLT with rates in Wasserstein distance and a conditional functional CLT with a rate in an integral distance over Barbour's class of smooth test functions. Additionally, we adapt Tikhomirov's method to obtain a bound in Kolmogorov distance for the conditional CLT.

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 paper establishes a functional correlation bound (FCB) with exponential decay for random Lasota-Yorke maps with holes. It derives this by first proving uniform Lasota-Yorke inequalities for the random family of open maps that control both the expanding dynamics and the mass loss through holes via a uniform lower bound on expansion and controlled distortion. These yield a transfer-operator contraction on a suitable Banach space, producing the FCB uniform in the random parameter. The FCB is then combined with abstract normal-approximation theorems from LNN25 and LS20 to obtain a conditional CLT with rates in Wasserstein distance, a conditional functional CLT with a rate in an integral distance over Barbour's class of smooth test functions, and a Kolmogorov-distance bound obtained by adapting Tikhomirov's method.

Significance. If the derivations hold, the result is significant because it successfully extends the functional correlation bound framework from closed systems to the random open setting, providing explicit exponential decay that is uniform over the randomness. This enables conditional limit theorems with rates for transient dynamics modeled by systems with holes, which are relevant in applications involving escape or absorption. The direct reduction of the CLT statements to the hypotheses of the cited abstract theorems, together with the uniform control on escaping trajectories, strengthens the applicability of correlation bounds in open random systems.

minor comments (3)
  1. [Theorem 2.1] The dependence of the constants in the uniform Lasota-Yorke inequalities on the size and location of the holes should be stated more explicitly in the statement of the main theorem to clarify the range of applicability.
  2. [Section 4] In the application of the abstract results from LNN25 and LS20, the precise verification that the FCB satisfies all required hypotheses (including the form of the test functions) could be expanded with a short checklist or table for reader convenience.
  3. A few instances of inconsistent notation appear when switching between the random parameter and the hole indicator function; a uniform symbol convention would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of the functional correlation bound for random Lasota-Yorke maps with holes and its use in conditional normal approximations. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no points requiring point-by-point responses at this stage. We remain available to incorporate any minor suggestions from the editor.

Circularity Check

0 steps flagged

No significant circularity; FCB derived independently from Lasota-Yorke inequalities

full rationale

The paper first derives uniform Lasota-Yorke inequalities for the random open maps, yielding a transfer-operator contraction on a Banach space that produces the functional correlation bound with exponential decay. This FCB is then fed into the hypotheses of the externally cited abstract normal-approximation theorems from LNN25 and LS20 to obtain the CLT statements. No step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the central FCB construction is self-contained and independent of the target CLT results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard properties of Lasota-Yorke maps and prior abstract normal-approximation theorems; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Lasota-Yorke type inequalities can be established for the random open maps to control escaping trajectories
    Invoked to extend FCB from closed to open systems.

pith-pipeline@v0.9.0 · 5477 in / 1373 out tokens · 56165 ms · 2026-05-10T06:21:14.812208+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Arras, G

    B. Arras, G. Mijoule, G. Poly, and Y. Swan. A new approach to the Stein-Tikhomirov method: with applications to the second Wiener chaos and Dickman convergence, 2016. Preprint, arXiv:1605.06819

    work page arXiv 2016

  2. [2]

    Atnip, G

    J. Atnip, G. Froyland, C. Gonz´ alez-Tokman, Y. Nakano, and S. Vaienti. Conditional limit theorems in dynamical systems. In preparation

    work page

  3. [3]

    Atnip, G

    J. Atnip, G. Froyland, C. Gonz´ alez-Tokman, and S. Vaienti. Thermodynamic formalism and perturbation formulae for quenched random open dynamical systems, 2023. Preprint, arXiv:2307.00774

    work page arXiv 2023

  4. [4]

    A. D. Barbour. Stein’s method for diffusion approximations.Probab. Theory Related Fields, 84(3):297–322, 1990. doi: 10.1007/BF01197887

    work page doi:10.1007/bf01197887 1990

  5. [5]

    A. D. Barbour, N. Ross, and G. Zheng. Stein’s method, smoothing and functional ap- proximation.Electron. J. Probab., 29:Paper No. 20, 29, 2024. doi: 10.1214/24-EJP1081

    work page doi:10.1214/24-ejp1081 2024

  6. [6]

    Collet, S

    P. Collet, S. Mart´ ınez, and V. Maume-Deschamps. On the existence of conditionally invariant probability measures in dynamical systems.Nonlinearity, 13(4):1263–1274,

    work page

  7. [7]

    doi: 10.1088/0951-7715/13/4/315

    work page doi:10.1088/0951-7715/13/4/315

  8. [8]

    M. F. Demers and C. Liverani. Central limit theorem for sequential dynamical systems,

    work page

  9. [9]

    Preprint, arXiv:2502.07765

    work page arXiv

  10. [10]

    Denker, M

    M. Denker, M. Gordin, and A. Sharova. A Poisson limit theorem for toral automor- phisms.Illinois J. Math., 48(1):1–20, 2004. doi: 10.1215/ijm/1258136170

    work page doi:10.1215/ijm/1258136170 2004

  11. [11]

    Dragiˇ cevi´ c and Y

    D. Dragiˇ cevi´ c and Y. Hafouta. Limit theorems for random expanding or Anosov dynam- ical systems and vector-valued observables.Ann. Henri Poincar´ e, 21(12):3869–3917,

    work page

  12. [12]

    doi: 10.1007/s00023-020-00965-7

    work page doi:10.1007/s00023-020-00965-7

  13. [13]

    Dragiˇ cevi´ c, G

    D. Dragiˇ cevi´ c, G. Froyland, C. Gonz´ alez-Tokman, and S. Vaienti. A spectral approach for quenched limit theorems for random expanding dynamical systems.Comm. Math. Phys., 360(3):1121–1187, 2018. doi: 10.1007/s00220-018-3158-4

    work page doi:10.1007/s00220-018-3158-4 2018

  14. [14]

    Gordin and M

    M. Gordin and M. Denker. The Poisson limit for automorphisms of two-dimensional tori driven by continued fractions.J. Math. Sci., 199(2):139–149, 2014. doi: 10.1007/s10958- 014-1841-z. FCB FOR OPEN DYNAMICS 41

    work page doi:10.1007/s10958- 2014

  15. [15]

    M. I. Gordin. A homoclinic version of the central limit theorem.J. Math. Sci., 68(4): 451–458, 1994. doi: 10.1007/BF01254269

    work page doi:10.1007/bf01254269 1994

  16. [16]

    Y. Hafouta. Explicit conditions for the CLT and related results for non-uniformly par- tially expanding random dynamical systems via effective RPF rates.Adv. Math., 426: Paper No. 109109, 86, 2023. doi: 10.1016/j.aim.2023.109109

    work page doi:10.1016/j.aim.2023.109109 2023

  17. [17]

    Hafouta and Y

    Y. Hafouta and Y. Kifer.Nonconventional Limit Theorems and Random Dynamics. World Scientific, 2018. doi: 10.1142/10849

    work page doi:10.1142/10849 2018

  18. [18]

    Haydn and F

    N. Haydn and F. Yang. Entry times distribution for mixing systems.J. Stat. Phys., 163 (2):374–392, 2016. doi: 10.1007/s10955-016-1487-y

    work page doi:10.1007/s10955-016-1487-y 2016

  19. [19]

    N. T. A. Haydn. Entry and return times distribution.Dynam. Systems, 28(3):333–353,

    work page

  20. [20]

    doi: 10.1080/14689367.2013.822459

    work page doi:10.1080/14689367.2013.822459 2013

  21. [21]

    O. Hella. Central limit theorems with a rate of convergence for sequences of transfor- mations, 2018. Preprint, arXiv:1811.06062

    work page arXiv 2018

  22. [22]

    Hella and J

    O. Hella and J. Lepp¨ anen. Central limit theorems with a rate of convergence for time-dependent intermittent maps.Stoch. Dyn., 20(4):2050025, 2020. doi: 10.1142/S0219493720500252

    work page doi:10.1142/s0219493720500252 2020

  23. [23]

    Hella, J

    O. Hella, J. Lepp¨ anen, and M. Stenlund. Stein’s method of normal approximation for dynamical systems.Stoch. Dyn., 20(4):2050021, 2020. doi: 10.1142/S0219493720500215

    work page doi:10.1142/s0219493720500215 2020

  24. [24]

    J. L. King. On M. Gordin’s homoclinic question.Internat. Math. Res. Notices, (5): 203–212, 1997. doi: 10.1155/S1073792897000147

    work page doi:10.1155/s1073792897000147 1997

  25. [25]

    Lepp¨ anen

    J. Lepp¨ anen. Functional correlation decay and multivariate normal approximation for non-uniformly expanding maps.Nonlinearity, 30(11):4239–4259, 2017. doi: 10.1088/1361-6544/aa85d0

    work page doi:10.1088/1361-6544/aa85d0 2017

  26. [26]

    Lepp¨ anen and M

    J. Lepp¨ anen and M. Stenlund. Sunklodas’ approach to normal approximation for time-dependent dynamical systems.J. Stat. Phys., 181(5):1523–1564, 2020. doi: 10.1007/s10955-020-02636-7

    work page doi:10.1007/s10955-020-02636-7 2020

  27. [27]

    Lepp¨ anen, Y

    J. Lepp¨ anen, Y. Nakajima, and Y. Nakano. Error bounds in a smooth metric for Brow- nian approximation of dynamical systems via Stein’s method.J. Stat. Phys., 192(12): Paper No. 168, 45, 2025. doi: 10.1007/s10955-025-03547-1

    work page doi:10.1007/s10955-025-03547-1 2025

  28. [28]

    Z. Liu, B. Saussol, S. Vaienti, and Z. Wang. Quenched invariance principle with a rate for random dynamical systems, 2025. Preprint, arXiv:2506.13167

    work page arXiv 2025

  29. [29]

    Liverani and V

    C. Liverani and V. Maume-Deschamps. Lasota–Yorke maps with holes: condition- ally invariant probability measures and invariant probability measures on the survivor set.Ann. Inst. H. Poincar´ e Probab. Statist., 39(3):385–412, 2003. doi: 10.1016/S0246- 0203(02)00005-5

    work page doi:10.1016/s0246- 2003

  30. [30]

    Psiloyenis.Mixing conditions and return times on Markov tow- ers

    Y. Psiloyenis.Mixing conditions and return times on Markov tow- ers. PhD thesis, University of Southern California, 2008. URL http://search.proquest.com/docview/304461750/. ProQuest LLC, Ann Arbor, MI

    work page arXiv 2008

  31. [31]

    C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables.Proc. Sixth Berkeley Sympos. Math. Statist. Probab., 2: 583–602, 1972

    work page 1972

  32. [32]

    Sunklodas

    J. Sunklodas. Rate of convergence in the central limit theorem for random variables with strong mixing.Lith. Math. J., 24(2):182–190, 1984

    work page 1984

  33. [33]

    A. N. Tihomirov. Convergence rate in the central limit theorem for weakly dependent random variables.Theory Probab. Appl., 25(4):790–809, 1980. doi: 10.1137/1125092. 42 JUHO LEPP ¨ANEN, YUTO NAKAJIMA, AND YUSHI NAKANO Department of Mathematics, Tokai University, Kanagawa, 259-1292, Japan Email address:leppanen.juho.heikki.g@tokai.ac.jp F aculty of Science...

    work page doi:10.1137/1125092 1980