Finite-order exponential mixing does not imply the central limit theorem

Abstract

We show that finite-order exponential mixing does not imply the central limit theorem for smooth diffeomorphisms. For every integer R≥2, we construct a 𝒞^∞ diffeomorphism T:M→ M of a compact smooth manifold M and a T-invariant probability measure μ such that (T,μ) is exponentially mixing of order R for 𝒞¹(M) observables, but not of order 2R+2. For some centred observable f₀∈𝒞^∞(M), no subsequence of the normalised sums 1/√n∑_i=0ⁿ⁻¹f₀∘ Tⁱ converges in distribution to a Gaussian law. In particular, exponential decay of two-point correlations implies neither exponential mixing of all orders nor the central limit theorem. We realise (T,μ) in two ways. In the first, T is a fixed linear Anosov diffeomorphism of 𝕋² and μ is singular with respect to Lebesgue measure. In the second, M can be any compact smooth surface without boundary and μ is a physical measure whose statistical basin contains a nonempty open set. The main theorems have been formalised in Lean 4.

Can we further improve the central limit theorem for stationary ρ-mixing sequences ?
Muneya Matsui
arXiv4 Sep 2026PreprintCC BY 4.0Full textProbability StatisticsarXiv
A Berry--Esseen Bound of Order (1+log n)/√n for Peng's Central Limit Theorem
Yongsheng Song
arXiv11 Oct 2026PreprintCC BY 4.0Full textProbabilityarXiv
Central limit theorem for Wasserstein projection - the case of convex order
Yuanlong Ruan
arXiv30 Aug 2026PreprintarXiv licenceFull textOptimisation & control ProbabilityarXiv
A Finite-Entropy Criterion for the Entropic Conditional Central Limit Theorem
Tong Ye, Liu-Quan Yao, Shuai Yuan +1
arXiv31 Aug 2026PreprintCC BY 4.0Full textProbabilityarXiv
Exponential Mixing for the Randomly Forced NLS Equation
Yuxuan Chen, Shengquan Xiang, Zhifei Zhang +1
Communications in Mathematical Physics28 Aug 2026Peer-reviewedLicence not statedFull text10 citedMathematical physicsOpenAlex

Cite

BibTeX RIS

BibTeX

@misc{cornudella2026finiteorder,
  title = {{Finite-order exponential mixing does not imply the central limit theorem}},
  author = {Bernat Bassols Cornudella and Matheus M Castro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17162},
  eprint = {2610.17162},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17162},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Finite-order exponential mixing does not imply the central limit theorem
AU  - Bernat Bassols Cornudella
AU  - Matheus M Castro
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17162
UR  - https://arxiv.org/abs/2610.17162
AB  - We show that finite-order exponential mixing does not imply the central limit theorem for smooth diffeomorphisms. For every integer R≥2, we construct a 𝒞^∞ diffeomorphism T:M→ M of a compact smooth manifold M and a T-invariant probability measure μ such that (T,μ) is exponentially mixing of order R for 𝒞^1(M) observables, but not of order 2R+2. For some centred observable f_0∈𝒞^∞(M), no subsequence of the normalised sums 1/√n∑_i=0^n−1f_0∘ T^i converges in distribution to a Gaussian law. In particular, exponential decay of two-point correlations implies neither exponential mixing of all orders nor the central limit theorem. We realise (T,μ) in two ways. In the first, T is a fixed linear Anosov diffeomorphism of 𝕋^2 and μ is singular with respect to Lebesgue measure. In the second, M can be any compact smooth surface without boundary and μ is a physical measure whose statistical basin contains a nonempty open set. The main theorems have been formalised in Lean 4.
ER  -

Abstract

Related

Cite