Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems

Mark Holland; Mike Todd

doi:10.1017/etds.2017.56

Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems

Mark Holland
, Mike Todd

Pure Mathematics

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

Abstract

For a measure-preserving dynamical system (X, ƒ, μ), we consider the time series of maxima M_n = max{X₁,…,X_n} associated to the process Xn = φ (ƒ^n-1(x)) generated by the dynamical system for some observable φ : Χ → R . Using a point-process approach we establish weak convergence of the process Y_n(t) = a_n(M_[nt] - b_n) to an extremal Y(t) process for suitable scaling constants a_n, b_n ∈ R . Convergence here takes place in the Skorokhod space D(0, ∞) with the J₁ topology. We also establish distributional results for the record times and record values of the corresponding maxima process.

Original language	English
Pages (from-to)	980-1001
Journal	Ergodic Theory and Dynamical Systems
Volume	39
Issue number	4
Early online date	7 Sept 2017
DOIs	https://doi.org/10.1017/etds.2017.56
Publication status	Published - Apr 2019

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© Cambridge University Press, 2017. This work has been made available online in accordance with the publisher’s policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1017/etds.2017.56
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title = "Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems",

abstract = "For a measure-preserving dynamical system (X, ƒ, μ), we consider the time series of maxima Mn = max\{X1,…,Xn\} associated to the process Xn = φ (ƒn-1(x)) generated by the dynamical system for some observable φ : Χ → R . Using a point-process approach we establish weak convergence of the process Yn(t) = an(M[nt] - bn) to an extremal Y(t) process for suitable scaling constants an, bn ∈ R . Convergence here takes place in the Skorokhod space D(0, ∞) with the J1 topology. We also establish distributional results for the record times and record values of the corresponding maxima process.",

author = "Mark Holland and Mike Todd",

note = "This research was partially supported by the London Mathematics Society (Scheme 4, no. 41126), and both authors thank the Erwin Schroedigner Institute (ESI) in Vienna were part of this work was carried out. MH wishes to thank the Department of Mathematics, University of Houston for hospitality and financial support, and MT thanks Exeter University for their hospitality and support.",

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AU - Todd, Mike

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N2 - For a measure-preserving dynamical system (X, ƒ, μ), we consider the time series of maxima Mn = max{X1,…,Xn} associated to the process Xn = φ (ƒn-1(x)) generated by the dynamical system for some observable φ : Χ → R . Using a point-process approach we establish weak convergence of the process Yn(t) = an(M[nt] - bn) to an extremal Y(t) process for suitable scaling constants an, bn ∈ R . Convergence here takes place in the Skorokhod space D(0, ∞) with the J1 topology. We also establish distributional results for the record times and record values of the corresponding maxima process.

AB - For a measure-preserving dynamical system (X, ƒ, μ), we consider the time series of maxima Mn = max{X1,…,Xn} associated to the process Xn = φ (ƒn-1(x)) generated by the dynamical system for some observable φ : Χ → R . Using a point-process approach we establish weak convergence of the process Yn(t) = an(M[nt] - bn) to an extremal Y(t) process for suitable scaling constants an, bn ∈ R . Convergence here takes place in the Skorokhod space D(0, ∞) with the J1 topology. We also establish distributional results for the record times and record values of the corresponding maxima process.

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DO - 10.1017/etds.2017.56

M3 - Article

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JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 4

ER -

Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems

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