scholarly journals Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems

2015 ◽  
Vol 51 (2) ◽  
pp. 545-556 ◽  
Author(s):  
Ian Melbourne ◽  
Roland Zweimüller
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Lévy Processes ◽  
Levy Processes ◽  
Hyperbolic Dynamical Systems ◽  
Nonuniformly Hyperbolic

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

2017 ◽  
Vol 39 (4) ◽  
pp. 980-1001
Author(s):  
MARK HOLLAND ◽  
MIKE TODD
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Dynamical Systems ◽  
Weak Convergence ◽  
Record Values ◽  
Process Approach ◽  
Record Times ◽  
Skorokhod Space ◽  
Extremal Process ◽  
Hyperbolic Dynamical Systems

For a measure-preserving dynamical system $({\mathcal{X}},f,\unicode[STIX]{x1D707})$, we consider the time series of maxima $M_{n}=\max \{X_{1},\ldots ,X_{n}\}$ associated to the process $X_{n}=\unicode[STIX]{x1D719}(f^{n-1}(x))$ generated by the dynamical system for some observable $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow \mathbb{R}$. Using a point-process approach we establish weak convergence of the process $Y_{n}(t)=a_{n}(M_{[nt]}-b_{n})$ to an extremal process $Y(t)$ for suitable scaling constants $a_{n},b_{n}\in \mathbb{R}$. Convergence here takes place in the Skorokhod space $\mathbb{D}(0,\infty )$ with the $J_{1}$ topology. We also establish distributional results for the record times and record values of the corresponding maxima process.

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