Weak Convergence to Self-Affine Processes in Dynamical Systems

1993 ◽  
pp. 255-262
Author(s):  
Michael Lacey
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Affine Processes

Weak convergence of dynamical systems in two timescales

2020 ◽  
Vol 142 ◽  
pp. 104718
Author(s):  
Gopal K. Basak ◽  
Amites Dasgupta
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence

scholarly journals Intermittent quasistatic dynamical systems: weak convergence of fluctuations

2018 ◽  
Vol 5 (1) ◽  
pp. 8-34 ◽  
Author(s):  
Juho Leppänen
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Time Evolution ◽  
Martingale Problem ◽  
Statistical Properties ◽  
Time Dependent ◽  
Stochastic Diffusion ◽  
Interval Maps ◽  
Well Posed ◽  
Over Time

Abstract This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influences. We focus on the case where the time-evolution is described by intermittent interval maps (Pomeau-Manneville maps) with time-dependent parameters. In a suitable range of parameters, we obtain a description of the statistical properties as a stochastic diffusion, by solving a well-posed martingale problem. The results extend those of a related recent study due to Dobbs and Stenlund, which concerned the case of quasistatic (uniformly) expanding systems.

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.

WEAK CONVERGENCE TO LÉVY STABLE PROCESSES IN DYNAMICAL SYSTEMS

2010 ◽  
Vol 10 (02) ◽  
pp. 263-289 ◽  
Author(s):  
MARTA TYRAN-KAMIŃSKA
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Stable Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stable Processes ◽  
Skorohod Space ◽  
Necessary And Sufficient

We study convergence of normalized ergodic sum processes to Lévy stable process in the Skorohod space with J1-topology. Our necessary and sufficient conditions allow us to prove or disprove such convergence for specific examples.

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