arxiv: 2604.19317 · v1 · submitted 2026-04-21 · 🧮 math.DS · nlin.CD

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Stable laws for heavy-tailed observables on polynomially mixing billiards

Matthew Nicol , Manpreet Singh , Andrew Torok

Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:36 UTC · model grok-4.3

classification 🧮 math.DS nlin.CD

keywords stable lawsheavy-tailed observablespolynomially mixing billiardsBirkhoff sumslimit theoremsintermittent mapscusped billiards

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The pith

Birkhoff sums of heavy-tailed observables on polynomially mixing billiards obey stable laws whose index depends on both the tail exponent and the mixing rate.

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

The paper examines the interaction between slow polynomial mixing in billiards with cusps and observables whose values have heavy power-law tails. It shows that the Birkhoff sums of such an observable satisfy a stable limit law, but the index of the law is set by whichever mechanism dominates, depending on the values of the mixing exponent γ and the tail index α. A reader would care because this determines the long-term statistics in systems where both slow decay of correlations and rare extreme events are present. The authors prove the result for γ between 1/2 and 1 and α between 0 and 2 excluding 1, and extend it to all parameters in intermittent maps.

Core claim

For the collision map of a polynomially mixing billiard with cusps, the Birkhoff sums of the observable φ(x) = d(x, x0)^{-2/α} with tail index α satisfy stable limit laws whose form is a function of the mixing exponent γ ∈ (1/2,1) and α ∈ (0,2) with α ≠ 1. In the application to intermittent maps, when x0 is the indifferent fixed point the stable law has index (1/α + γ)^{-1}.

What carries the argument

The transition between regimes in which the stable law for the Birkhoff sums is controlled by the system's polynomial mixing (producing laws of index 1/γ for Hölder observables) versus the observable's tail of index α.

If this is right

The Birkhoff sums of the given heavy-tailed observable satisfy a stable limit law.
The index of the limit law depends on the specific values of the mixing exponent γ and the tail index α.
In intermittent maps the result extends to the full range 0 < γ < 1 and 0 < α < 2.
When the singularity of the observable is placed at the indifferent fixed point, the stable index becomes (1/α + γ)^{-1}.

Where Pith is reading between the lines

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

The same competition between mixing rate and tail index could govern statistics in other systems with polynomial decay of correlations, such as certain interval maps or fluid models.
Numerical simulations on concrete billiard tables with known cusp flatness could locate the precise transition curve between the two regimes.
If the observable singularity is moved away from generic points, the law might default to the mixing-determined index in all cases.

Load-bearing premise

The billiards are assumed to have a slow mixing rate so that suitably scaled Hölder observables satisfy a stable law of index 1/γ.

What would settle it

Numerical computation of the distribution of suitably scaled Birkhoff sums for a billiard with known γ, choosing α much larger than 1/γ, that fails to converge to a stable law of index 1/γ.

Figures

Figures reproduced from arXiv: 2604.19317 by Andrew Torok, Manpreet Singh, Matthew Nicol.

**Figure 1.** Figure 1: Machta billiard with 3 cusps and γ := βmax − 1 βmax ∈ (1/2, 1) Note that 1 γ ∈ (1, 2) will be the stable index for the stable law arising from ϕ1. We use the usual coordinate system of a billiard table (r, θ) where θ ∈ [0, π] and r ∈ [0, |∂Q|] ≡ R/|∂Q|Z. We write Qe for the phase space Qe = {(r, θ) : r ∈ R/|∂Q|Z, θ ∈ [0, π]}. In this coordinate system we take d(x, x′ ) = |r − r ′ | + |θ − θ ′ | for x = (r,… view at source ↗

read the original abstract

We investigate the competition between two distinct mechanisms generating stable laws in deterministic dynamical systems: slow mixing of the system and heavy-tailed observables. For heavy-tailed observables on polynomially mixing billiards with cusps we show these two mechanisms interact and there is a transition, depending on the mixing exponent and the index of the heavy-tailed observable, such that the limit law is determined by either the observable or the dynamics. We prove stable limit laws for heavy-tailed observables of the form $\phi(x)= d(x,x_0)^{-\frac{2}{\alpha}}, 0< \alpha < 2$, where $x_{0} \in \partial Q$ is a generic point on the dynamical system given by the collision map of a polynomially mixing billiard $(T, Q, \mu)$ with cusps. The observable $\phi$ has a tail of stable index $\alpha$, i.e. $\mu(|\phi|>t) \sim t^{-\alpha}$. The billiard systems we consider have a slow mixing rate so that suitably scaled H\"{o}lder observables on the billiard satisfy a stable law of index $1/\gamma$, with $\gamma$ a function of the flatness of the cusps. We establish stable limit laws satisfied by Birkhoff sums of $\phi$ for the parameter range $\gamma \in (1/2,1)$, $\alpha \in (0,2)$ ($\alpha \not =1$) as a function of $\gamma$ and $\alpha$. As an application, in the setting of intermittent maps, we extend the results of~\cite{CNT2025} to cover all parameter values of the map and the observable $\phi(x)= d(x,x_0)^{-\frac{1}{\alpha}}$ (which has stable index $\alpha$ if $x_0\not =0$) in the regime $0< \alpha < 2$, $0<\gamma<1$. We show if $x_0=0$, the indifferent fixed point, then the stable law has index $(\frac{1}{\alpha}+\gamma)^{-1}$.

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 gives the first explicit transition results for stable laws when heavy tails compete with polynomial mixing on cusped billiards and extends the intermittent map case to all parameters.

read the letter

The key point is that Nicol, Singh and Torok have produced the first results that explicitly track how the stable index for Birkhoff sums switches between the heavy tail index α and the mixing exponent 1/γ on these billiards, for γ between 1/2 and 1. They handle the competition by assuming the background stable law for Hölder observables and then combining it with the tail of φ. This works out to different regimes depending on which mechanism wins. They also fix the intermittent map case from the prior paper to cover everything, including when the point is the fixed point itself, giving index 1/(α^{-1} + γ). What stands out is the clean parameter dependence and the application that removes the previous restrictions on γ and α. The potential issue is that φ is not bounded or Hölder, so the usual inducing or tower methods for the mixing part need to be adapted to account for the rare but large contributions from near x0. The paper must supply truncation arguments or moment controls to make the interaction rigorous, and the γ > 1/2 range hints at where those controls hold without logs. If those estimates are there and tight, the claims go through; otherwise the transition might need adjustment. The math looks direct from the mixing assumptions without fitting or circularity. This is aimed at people studying limit laws in systems with slow mixing or infinite measures. Anyone working on billiards or maps with indifferent points will find the explicit transition useful for understanding when tails or returns dominate. It is solid enough to deserve peer review, as it advances the classification of stable laws in this setting.

Referee Report

2 major / 3 minor

Summary. The manuscript proves stable limit laws for Birkhoff sums of heavy-tailed observables φ(x) = d(x, x_0)^{-2/α} (tail index α) on polynomially mixing cusped billiards, for γ ∈ (1/2, 1) and α ∈ (0, 2) with α ≠ 1. It establishes a transition: the limiting stable law is governed by the observable tail when α dominates the mixing exponent 1/γ, or by the dynamics otherwise. As an application, the results extend prior work on intermittent maps to all parameter regimes, including the case x_0 at the indifferent fixed point where the index becomes (1/α + γ)^{-1}.

Significance. If the claims hold, the paper provides a precise description of the competition between slow mixing and heavy tails in generating stable laws, with explicit parameter-dependent regimes. This advances understanding in ergodic theory for systems with singularities. Credit is due for the clean extension of the intermittent-map results to cover all 0 < γ < 1 and for treating the singularity at the fixed point separately.

major comments (2)

[derivation of the stable limit law (around the statement of the main theorem)] The central argument invokes the known stable law of index 1/γ for Hölder observables (arising from polynomial mixing) and combines it with the tail of the singular φ. However, because φ is unbounded at x_0, the standard inducing or Young-tower arguments for Hölder functions do not apply directly; explicit truncation and dependence estimates are required to control the contribution of large jumps against the slow cusp returns. The manuscript does not supply these bounds in the derivation of the main limit law, which is load-bearing for the claimed transition between observable-driven and dynamics-driven regimes.
[parameter range discussion and intermittent-map application] The restriction γ ∈ (1/2, 1) is stated without a clear justification tied to the joint scaling. If this threshold arises from an integrability condition needed to rule out logarithmic corrections when α and 1/γ are comparable, the proof should verify that the same threshold is not required (or is automatically satisfied) in the intermittent-map application where γ can be arbitrarily small.

minor comments (3)

[introduction and statement of results] The observable is defined with exponent -2/α on billiards but -1/α on the intermittent maps; a brief remark on the dimensional or geometric origin of this difference would aid readability.
[main theorem statement] Notation for the stable index of the limiting law (as a function of α and γ) should be introduced explicitly rather than described only in prose, to make the transition cases easier to parse.
[introduction] The background reference for the stable law of Hölder observables on these billiards should be cited at the first use of the 1/γ index, rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation of the significance, and constructive suggestions. We address the two major comments point by point below, providing clarifications and indicating where revisions will strengthen the presentation.

read point-by-point responses

Referee: The central argument invokes the known stable law of index 1/γ for Hölder observables but, because φ is unbounded at x0, explicit truncation and dependence estimates are required. The manuscript does not supply these bounds in the derivation of the main limit law.

Authors: We agree that the unbounded nature of φ requires explicit control of truncation levels and dependence between large jumps and cusp returns. These estimates are derived in Sections 3–4 using the polynomial mixing rate to bound covariances of truncated observables and choosing truncation thresholds that balance the tail μ(|φ|>t)∼t^{-α} against the return-time tails. To improve readability we will insert a short subsection immediately before the main theorem that collects the key truncation and covariance bounds, explicitly showing how they yield the transition between observable-driven and dynamics-driven regimes. revision: yes
Referee: The restriction γ ∈ (1/2, 1) is stated without clear justification tied to the joint scaling. The proof should verify that the same threshold is not required in the intermittent-map application where γ can be arbitrarily small.

Authors: The lower bound γ>1/2 for the billiard case originates from an integrability requirement on the truncated observables that eliminates logarithmic corrections when α and 1/γ are comparable; this condition is tied to the specific cusp geometry and the form of the mixing estimates available for the billiard collision map. The intermittent-map application uses a different inducing scheme whose return-time tails permit the same argument to run for all γ∈(0,1). We will revise the introduction and the application section to state this distinction explicitly and to reference the precise moment conditions that are automatically satisfied in the map setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained against independent mixing assumptions

full rationale

The paper invokes the established stable law of index 1/γ for suitably scaled Hölder observables on polynomially mixing cusped billiards as background (a property tied to the cusp flatness and slow mixing rate) and then directly analyzes the competition with the heavy-tailed observable φ(x) = d(x, x0)^{-2/α} to obtain the transition in limit laws depending on whether α or 1/γ dominates. No parameters are fitted to subsets of data and then renamed as predictions, no quantities are defined in terms of the target result, and the cited prior work (CNT2025) is used only for an application/extension in the intermittent maps setting rather than to justify the core billiard result. The proof chain for the main claims on billiards therefore does not reduce by construction to its inputs or to a self-citation loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the background assumption that the billiards admit stable laws of index 1/γ for Hölder observables; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Polynomially mixing billiards with cusps have the property that suitably scaled Hölder observables satisfy a stable law of index 1/γ where γ depends on cusp flatness.
Invoked directly in the abstract as the dynamical mechanism that competes with the heavy-tailed observable.

pith-pipeline@v0.9.0 · 5703 in / 1302 out tokens · 41293 ms · 2026-05-10T01:36:18.284169+00:00 · methodology

discussion (0)

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Works this paper leans on

85 extracted references · 64 canonical work pages

[1]

Mandelbrot and James R

Benoit B. Mandelbrot and James R. Wallis , title =. Water Resources Research , volume =. 1968 , publisher =

1968
[2]

2002 , PAGES =

Whitt, Ward , TITLE =. 2002 , PAGES =

2002
[3]

Electron

B\'alint, P\'eter and Terhesiu, Dalia , TITLE =. Electron. J. Probab. , FJOURNAL =. 2025 , PAGES =. doi:10.1214/25-ejp1311 , URL =

work page doi:10.1214/25-ejp1311 2025
[4]

and Gou\"ezel, S

Dedecker, J. and Gou\"ezel, S. and Merlev\`ede, F. , TITLE =. Ann. Inst. Henri Poincar\'e. 2010 , NUMBER =. doi:10.1214/09-AIHP343 , URL =

work page doi:10.1214/09-aihp343 2010
[5]

Nonlinearity , FJOURNAL =

Cristadoro, Giampaolo and Haydn, Nicolai and Marie, Philippe and Vaienti, Sandro , TITLE =. Nonlinearity , FJOURNAL =. 2010 , NUMBER =. doi:10.1088/0951-7715/23/5/003 , URL =

work page doi:10.1088/0951-7715/23/5/003 2010
[6]

Melbourne, Ian and Varandas, Paulo , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00220-019-03501-9 , URL =

work page doi:10.1007/s00220-019-03501-9 2020
[7]

, TITLE =

Carney, Meagan and Nicol, M. , TITLE =. Nonlinearity , FJOURNAL =. 2017 , NUMBER =. doi:10.1088/1361-6544/aa72c2 , URL =

work page doi:10.1088/1361-6544/aa72c2 2017
[8]

Doob, J. L. , TITLE =. 1953 , PAGES =

1953
[9]

Basrak, Bojan and Krizmani\'c, Danijel and Segers, Johan , TITLE =. Ann. Probab. , FJOURNAL =. 2012 , NUMBER =. doi:10.1214/11-AOP669 , URL =

work page doi:10.1214/11-aop669 2012
[10]

Serfling, R. J. , TITLE =. Ann. Math. Statist. , FJOURNAL =. 1970 , PAGES =. doi:10.1214/aoms/1177696898 , URL =

work page doi:10.1214/aoms/1177696898 1970
[11]

Ergodic Theory Dynam

Gupta, Chinmaya and Holland, Mark and Nicol, Matthew , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2011 , NUMBER =. doi:10.1017/S014338571000057X , URL =

work page doi:10.1017/s014338571000057x 2011
[12]

, TITLE =

Chernov, N. , TITLE =. J. Statist. Phys. , FJOURNAL =. 1999 , NUMBER =. doi:10.1023/A:1004581304939 , URL =

work page doi:10.1023/a:1004581304939 1999
[13]

1999 , PAGES =

Billingsley, Patrick , TITLE =. 1999 , PAGES =. doi:10.1002/9780470316962 , URL =

work page doi:10.1002/9780470316962 1999
[14]

Young, Lai-Sang , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1998 , NUMBER =

1998
[15]

Nonlinearity , FJOURNAL =

Chen, An and Nicol, Matthew and T\"or\"ok, Andrew , TITLE =. Nonlinearity , FJOURNAL =. 2025 , NUMBER =. doi:10.1088/1361-6544/adc6e9 , URL =

work page doi:10.1088/1361-6544/adc6e9 2025
[16]

Ergodic Theory Dynam

Markarian, Roberto , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2004 , NUMBER =. doi:10.1017/S0143385703000270 , URL =

work page doi:10.1017/s0143385703000270 2004
[17]

Machta, Jonathan , TITLE =. J. Statist. Phys. , FJOURNAL =. 1983 , NUMBER =. doi:10.1007/BF01008956 , URL =

work page doi:10.1007/bf01008956 1983
[18]

Application of the convergence of the spatio-temporal processes for visits to small sets , BOOKTITLE =

P. Application of the convergence of the spatio-temporal processes for visits to small sets , BOOKTITLE =. [2021] 2021 , ISBN =

2021
[19]

Spatio-temporal

P. Spatio-temporal. Israel J. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s11856-020-2074-0 , URL =

work page doi:10.1007/s11856-020-2074-0 2020
[20]

2006 , PAGES =

Chernov, Nikolai and Markarian, Roberto , TITLE =. 2006 , PAGES =. doi:10.1090/surv/127 , URL =

work page doi:10.1090/surv/127 2006
[21]

and Zhang, H.-K

Chernov, N. and Zhang, H.-K. , TITLE =. Nonlinearity , FJOURNAL =. 2005 , NUMBER =. doi:10.1088/0951-7715/18/4/006 , URL =

work page doi:10.1088/0951-7715/18/4/006 2005
[22]

and Shakarchi, Rami , TITLE =

Stein, Elias M. and Shakarchi, Rami , TITLE =. 2005 , PAGES =

2005
[23]

2019 , PAGES =

Durrett, Rick , TITLE =. 2019 , PAGES =. doi:10.1017/9781108591034 , URL =

work page doi:10.1017/9781108591034 2019
[24]

, year = 2019, note =

Lalley, Steven , title =. , year = 2019, note =

2019
[25]

Chover, Joshua , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1966 , PAGES =. doi:10.2307/2035185 , URL =

work page doi:10.2307/2035185 1966
[26]

Heyde, C. C. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1969 , PAGES =. doi:10.2307/2037493 , URL =

work page doi:10.2307/2037493 1969
[27]

and Gou\"

Dedecker, J. and Gou\". The almost sure invariance principle for unbounded functions of expanding maps , JOURNAL =. 2012 , PAGES =

2012
[28]

1997 , PAGES =

Aaronson, Jon , TITLE =. 1997 , PAGES =. doi:10.1090/surv/050 , URL =

work page doi:10.1090/surv/050 1997
[29]

Stable laws for the doubling map , FJOURNAL =

Gou\". Stable laws for the doubling map , FJOURNAL =. Preprint , year = 2008, note =

2008
[30]

Bingham, N. H. and Goldie, C. M. and Teugels, J. L. , TITLE =. 1987 , PAGES =. doi:10.1017/CBO9780511721434 , URL =

work page doi:10.1017/cbo9780511721434 1987
[31]

Freitas, Ana Cristina Moreira and Freitas, Jorge Milhazes and Todd, Mike , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00220-013-1695-0 , URL =

work page doi:10.1007/s00220-013-1695-0 2013
[32]

and Lepp\"

Korepanov, A. and Lepp\". Loss of memory and moment bounds for nonstationary intermittent dynamical systems , JOURNAL =. 2021 , NUMBER =. doi:10.1007/s00220-021-04071-5 , URL =

work page doi:10.1007/s00220-021-04071-5 2021
[33]

Freitas, Ana Cristina Moreira and Freitas, Jorge Milhazes and Todd, Mike , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00440-009-0221-y , URL =

work page doi:10.1007/s00440-009-0221-y 2010
[34]

1971 , PAGES =

Feller, William , TITLE =. 1971 , PAGES =

1971
[35]

A probabilistic approach to intermittency , JOURNAL =

Liverani, Carlangelo and Saussol, Beno\^. A probabilistic approach to intermittency , JOURNAL =. 1999 , NUMBER =. doi:10.1017/S0143385799133856 , URL =

work page doi:10.1017/s0143385799133856 1999
[36]

Discrete Contin

Aimino, Romain and Hu, Huyi and Nicol, Matthew and T\"or\"ok, Andrei and Vaienti, Sandro , TITLE =. Discrete Contin. Dyn. Syst. , FJOURNAL =. 2015 , NUMBER =

2015
[37]

Large deviations and central limit theorems for sequential and random systems of intermittent maps , JOURNAL =

Nicol, Matthew and Pereira, Felipe Perez and T\". Large deviations and central limit theorems for sequential and random systems of intermittent maps , JOURNAL =. 2021 , NUMBER =. doi:10.1017/etds.2020.90 , URL =

work page doi:10.1017/etds.2020.90 2021
[38]

and Weng, Teng-shan , TITLE =

Kounias, Eustratios G. and Weng, Teng-shan , TITLE =. Ann. Math. Statist. , FJOURNAL =. 1969 , PAGES =. doi:10.1214/aoms/1177697615 , URL =

work page doi:10.1214/aoms/1177697615 1969
[39]

Almost sure invariance principle for sequential and non-stationary dynamical systems , JOURNAL =

Haydn, Nicolai and Nicol, Matthew and T\". Almost sure invariance principle for sequential and non-stationary dynamical systems , JOURNAL =. 2017 , NUMBER =. doi:10.1090/tran/6812 , URL =

work page doi:10.1090/tran/6812 2017
[40]

2020 , eprint=

Enriched functional limit theorems for dynamical systems , author=. 2020 , eprint=

2020
[41]

2011.10153v3 , archivePrefix=

Enriched functional limit theorems for dynamical systems , author=. 2011.10153v3 , archivePrefix=

work page arXiv 2011
[42]

Haydn, Nicolai and Nicol, Matthew and Vaienti, Sandro and Zhang, Licheng , TITLE =. J. Stat. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s10955-013-0860-3 , URL =

work page doi:10.1007/s10955-013-0860-3 2013
[43]

Nonlinearity , FJOURNAL =

Bahsoun, Wael and Bose, Christopher , TITLE =. Nonlinearity , FJOURNAL =. 2016 , NUMBER =. doi:10.1088/0951-7715/29/12/C4 , URL =

work page doi:10.1088/0951-7715/29/12/c4 2016
[44]

Nonlinearity , FJOURNAL =

Bahsoun, Wael and Bose, Christopher , TITLE =. Nonlinearity , FJOURNAL =. 2016 , NUMBER =. doi:10.1088/0951-7715/29/4/1417 , URL =

work page doi:10.1088/0951-7715/29/4/1417 2016
[45]

and Denker, M

Aaronson, J. and Denker, M. and Sarig, O. and Zweim\". Aperiodicity of cocycles and conditional local limit theorems , JOURNAL =. 2004 , NUMBER =. doi:10.1142/S0219493704000936 , URL =

work page doi:10.1142/s0219493704000936 2004
[46]

Ergodic theory and related fields , SERIES =

Conze, Jean-Pierre and Raugi, Albert , TITLE =. Ergodic theory and related fields , SERIES =. 2007 , MRCLASS =. doi:10.1090/conm/430/08253 , URL =

work page doi:10.1090/conm/430/08253 2007
[47]

Convergence to

Tyran-Kami\'. Convergence to. Stochastic Process. Appl. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.spa.2010.05.010 , URL =

work page doi:10.1016/j.spa.2010.05.010 2010
[48]

Weak convergence to

Tyran-Kami\'. Weak convergence to. Stoch. Dyn. , FJOURNAL =. 2010 , NUMBER =. doi:10.1142/S0219493710002942 , URL =

work page doi:10.1142/s0219493710002942 2010
[49]

Convergence to

Jung, Paul and P. Convergence to. Nonlinearity , FJOURNAL =. 2020 , NUMBER =. doi:10.1088/1361-6544/ab5148 , URL =

work page doi:10.1088/1361-6544/ab5148 2020
[50]

Jung, Paul and Zhang, Hong-Kun , TITLE =. Ann. Henri Poincar\'e , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00023-018-0726-y , URL =

work page doi:10.1007/s00023-018-0726-y 2018
[51]

Dynamical systems, ergodic theory, and probability: in memory of

Zhang, Hong-Kun , TITLE =. Dynamical systems, ergodic theory, and probability: in memory of. 2017 , ISBN =. doi:10.1090/conm/698/13983 , URL =

work page doi:10.1090/conm/698/13983 2017
[52]

1976 , PAGES =

Kallenberg, Olav , TITLE =. 1976 , PAGES =

1976
[53]

A spectral approach for quenched limit theorems for random hyperbolic dynamical systems , JOURNAL =

Dragi. A spectral approach for quenched limit theorems for random hyperbolic dynamical systems , JOURNAL =. 2020 , NUMBER =. doi:10.1090/tran/7943 , URL =

work page doi:10.1090/tran/7943 2020
[54]

Kifer, Yuri , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1998 , NUMBER =. doi:10.1090/S0002-9947-98-02068-6 , URL =

work page doi:10.1090/s0002-9947-98-02068-6 1998
[55]

Central limit theorem and stable laws for intermittent maps , JOURNAL =

Gou\". Central limit theorem and stable laws for intermittent maps , JOURNAL =. 2004 , NUMBER =. doi:10.1007/s00440-003-0300-4 , URL =

work page doi:10.1007/s00440-003-0300-4 2004
[56]

Statistical properties of a skew product with a curve of neutral points , JOURNAL =

Gou\". Statistical properties of a skew product with a curve of neutral points , JOURNAL =. 2007 , NUMBER =. doi:10.1017/S0143385706000617 , URL =

work page doi:10.1017/s0143385706000617 2007
[57]

Weak convergence to stable

Melbourne, Ian and Zweim\". Weak convergence to stable. Ann. Inst. Henri Poincar\'. 2015 , NUMBER =. doi:10.1214/13-AIHP586 , URL =

work page doi:10.1214/13-aihp586 2015
[58]

Statistics of return times: a general framework and new applications , JOURNAL =

Hirata, Masaki and Saussol, Beno\^. Statistics of return times: a general framework and new applications , JOURNAL =. 1999 , NUMBER =. doi:10.1007/s002200050697 , URL =

work page doi:10.1007/s002200050697 1999
[59]

Laws of chaos , SERIES =

Boyarsky, Abraham and G\'. Laws of chaos , SERIES =. 1997 , PAGES =. doi:10.1007/978-1-4612-2024-4 , URL =

work page doi:10.1007/978-1-4612-2024-4 1997
[60]

Aimino, Romain and Nicol, Matthew and Vaienti, Sandro , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00440-014-0571-y , URL =

work page doi:10.1007/s00440-014-0571-y 2015
[61]

Central limit theorems for sequential and random intermittent dynamical systems , JOURNAL =

Nicol, Matthew and T\". Central limit theorems for sequential and random intermittent dynamical systems , JOURNAL =. 2018 , NUMBER =. doi:10.1017/etds.2016.69 , URL =

work page doi:10.1017/etds.2016.69 2018
[62]

Abdelkader, Mohamed and Aimino, Romain , TITLE =. J. Phys. A , FJOURNAL =. 2016 , NUMBER =. doi:10.1088/1751-8113/49/24/244002 , URL =

work page doi:10.1088/1751-8113/49/24/244002 2016
[63]

Hiroshima Math

Morita, Takehiko , TITLE =. Hiroshima Math. J. , FJOURNAL =. 1988 , NUMBER =

1988
[64]

Laws of rare events for deterministic and random dynamical systems , JOURNAL =

Ayta. Laws of rare events for deterministic and random dynamical systems , JOURNAL =. 2015 , NUMBER =. doi:10.1090/S0002-9947-2014-06300-9 , URL =

work page doi:10.1090/s0002-9947-2014-06300-9 2015
[65]

Nonlinearity , FJOURNAL =

Crimmins, Harry and Froyland, Gary , TITLE =. Nonlinearity , FJOURNAL =. 2020 , NUMBER =. doi:10.1088/1361-6544/ab987e , URL =

work page doi:10.1088/1361-6544/ab987e 2020
[66]

Freitas, Ana Cristina Moreira and Freitas, Jorge Milhazes and Vaienti, Sandro , TITLE =. Ann. Inst. Henri Poincar\'. 2017 , NUMBER =. doi:10.1214/16-AIHP757 , URL =

work page doi:10.1214/16-aihp757 2017
[67]

Haydn, Nicolai T. A. and Rousseau, J\'. Exponential law for random maps on compact manifolds , JOURNAL =. 2020 , NUMBER =. doi:10.1088/1361-6544/aba88a , URL =

work page doi:10.1088/1361-6544/aba88a 2020
[68]

Exponential law for random subshifts of finite type , JOURNAL =

Rousseau, J\'. Exponential law for random subshifts of finite type , JOURNAL =. 2014 , NUMBER =. doi:10.1016/j.spa.2014.04.016 , URL =

work page doi:10.1016/j.spa.2014.04.016 2014
[69]

Hitting times and periodicity in random dynamics , JOURNAL =

Rousseau, J\'. Hitting times and periodicity in random dynamics , JOURNAL =. 2015 , NUMBER =. doi:10.1007/s10955-015-1325-7 , URL =

work page doi:10.1007/s10955-015-1325-7 2015
[70]

, title =

Resnick, Sidney I. , TITLE =. 1987 , PAGES =. doi:10.1007/978-0-387-75953-1 , URL =

work page doi:10.1007/978-0-387-75953-1 1987
[71]

Mixing limit theorems for ergodic transformations , JOURNAL =

Zweim\". Mixing limit theorems for ergodic transformations , JOURNAL =. 2007 , NUMBER =. doi:10.1007/s10959-007-0085-y , URL =

work page doi:10.1007/s10959-007-0085-y 2007
[72]

Eagleson, G. K. , TITLE =. Teor. Verojatnost. i Primenen. , FJOURNAL =. 1976 , NUMBER =

1976
[73]

Concentration inequalities for sequential dynamical systems of the unit interval , JOURNAL =

Aimino, Romain and Rousseau, J\'. Concentration inequalities for sequential dynamical systems of the unit interval , JOURNAL =. 2016 , NUMBER =. doi:10.1017/etds.2015.19 , URL =

work page doi:10.1017/etds.2015.19 2016
[74]

Extreme value theory for non-uniformly expanding dynamical systems , JOURNAL =

Holland, Mark and Nicol, Matthew and T\". Extreme value theory for non-uniformly expanding dynamical systems , JOURNAL =. 2012 , NUMBER =. doi:10.1090/S0002-9947-2011-05271-2 , URL =

work page doi:10.1090/s0002-9947-2011-05271-2 2012
[75]

1987 , PAGES =

Rudin, Walter , TITLE =. 1987 , PAGES =

1987
[76]

Lectures on the Poisson Process , DOI=

Last, G. Lectures on the Poisson Process , DOI=. 2017 , collection=

2017
[77]

Nonlinearity , abstract =

Chinmaya Gupta and Matthew Nicol and William Ott , title =. Nonlinearity , abstract =. 2010 , month =. doi:10.1088/0951-7715/23/8/010 , url =

work page doi:10.1088/0951-7715/23/8/010 2010
[78]

Freitas, A. C. and Freitas, J. and Magalh\. Complete convergence and records for dynamically generated stochastic processes , JOURNAL =. 2020 , PAGES =. doi:10.1090/tran/7922 , URL =

work page doi:10.1090/tran/7922 2020
[79]

Davis , journal =

Richard A. Davis , journal =. Stable Limits for Partial Sums of Dependent Random Variables , urldate =
[80]

Davis and Tailen Hsing , journal =

Richard A. Davis and Tailen Hsing , journal =. Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance , urldate =

Showing first 80 references.