### Abstract

A dynamical array consists of a family of functions {f_{n,i}1 : 1 ≤ i ≤ k_{n}, n ≥ 1} and a family of initial times {τ_{n,i} : 1 ≤ i ≤ k_{n}, n ≥ 1}. For a dynamical system (X, T) we identify distributional limits for sums of the form S_{n} = 1/s_{n} ^{kn}_{i=1} [f_{n,i} T^{Tn,i} - a_{n,i}] n ≥ 1 for suitable (non-random) constants s_{n} > 0 and a_{n,i} ϵ ℝ. We derive a Lindeberg-type central limit theorem for dynamical arrays. Applications include new central limit theorems for functions which are not locally Lipschitz continuous and central limit theorems for statistical functions of time series obtained from GibbsMarkov systems. Our results, which hold for more general dynamics, are stated in the context of GibbsMarkov dynamical systems for convenience.

Original language | English (US) |
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Pages (from-to) | 4587-4613 |

Number of pages | 27 |

Journal | Nonlinearity |

Volume | 30 |

Issue number | 12 |

DOIs | |

State | Published - Nov 16 2017 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

### Cite this

*Nonlinearity*,

*30*(12), 4587-4613. https://doi.org/10.1088/1361-6544/aa8ca2

}

*Nonlinearity*, vol. 30, no. 12, pp. 4587-4613. https://doi.org/10.1088/1361-6544/aa8ca2

**Lindeberg theorem for Gibbs-Markov dynamics.** / Denker, Manfred Heinz; Senti, Samuel; Zhang, Xuan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Lindeberg theorem for Gibbs-Markov dynamics

AU - Denker, Manfred Heinz

AU - Senti, Samuel

AU - Zhang, Xuan

PY - 2017/11/16

Y1 - 2017/11/16

N2 - A dynamical array consists of a family of functions {fn,i1 : 1 ≤ i ≤ kn, n ≥ 1} and a family of initial times {τn,i : 1 ≤ i ≤ kn, n ≥ 1}. For a dynamical system (X, T) we identify distributional limits for sums of the form Sn = 1/sn kni=1 [fn,i TTn,i - an,i] n ≥ 1 for suitable (non-random) constants sn > 0 and an,i ϵ ℝ. We derive a Lindeberg-type central limit theorem for dynamical arrays. Applications include new central limit theorems for functions which are not locally Lipschitz continuous and central limit theorems for statistical functions of time series obtained from GibbsMarkov systems. Our results, which hold for more general dynamics, are stated in the context of GibbsMarkov dynamical systems for convenience.

AB - A dynamical array consists of a family of functions {fn,i1 : 1 ≤ i ≤ kn, n ≥ 1} and a family of initial times {τn,i : 1 ≤ i ≤ kn, n ≥ 1}. For a dynamical system (X, T) we identify distributional limits for sums of the form Sn = 1/sn kni=1 [fn,i TTn,i - an,i] n ≥ 1 for suitable (non-random) constants sn > 0 and an,i ϵ ℝ. We derive a Lindeberg-type central limit theorem for dynamical arrays. Applications include new central limit theorems for functions which are not locally Lipschitz continuous and central limit theorems for statistical functions of time series obtained from GibbsMarkov systems. Our results, which hold for more general dynamics, are stated in the context of GibbsMarkov dynamical systems for convenience.

UR - http://www.scopus.com/inward/record.url?scp=85036616132&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85036616132&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/aa8ca2

DO - 10.1088/1361-6544/aa8ca2

M3 - Article

VL - 30

SP - 4587

EP - 4613

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 12

ER -