Lindeberg theorem for Gibbs-Markov dynamics

Manfred Denker; Samuel Senti; Xuan Zhang

doi:10.1088/1361-6544/aa8ca2

Lindeberg theorem for Gibbs-Markov dynamics

Manfred Denker, Samuel Senti, Xuan Zhang

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

A dynamical array consists of a family of functions {f_n,i1 : 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)
Pages (from-to)	4587-4613
Number of pages	27
Journal	Nonlinearity
Volume	30
Issue number	12
DOIs	https://doi.org/10.1088/1361-6544/aa8ca2
State	Published - Nov 16 2017

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics
General Physics and Astronomy
Applied Mathematics

Access to Document

10.1088/1361-6544/aa8ca2

Cite this

@article{61f6f0327fdf4479afd913d86d358990,

title = "Lindeberg theorem for Gibbs-Markov dynamics",

abstract = "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.",

author = "Manfred Denker and Samuel Senti and Xuan Zhang",

note = "Publisher Copyright: {\textcopyright} 2017 IOP Publishing Ltd & London Mathematical Society.",

year = "2017",

month = nov,

day = "16",

doi = "10.1088/1361-6544/aa8ca2",

language = "English (US)",

volume = "30",

pages = "4587--4613",

journal = "Nonlinearity",

issn = "0951-7715",

publisher = "IOP Publishing Ltd.",

number = "12",

}

TY - JOUR

T1 - Lindeberg theorem for Gibbs-Markov dynamics

AU - Denker, Manfred

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

AN - SCOPUS:85036616132

SN - 0951-7715

VL - 30

SP - 4587

EP - 4613

JO - Nonlinearity

JF - Nonlinearity

IS - 12

ER -

Lindeberg theorem for Gibbs-Markov dynamics

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this