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Characterizations of ergodic stationary stable processes via the dynamical functional

  • Krzysztof Podgórski
  • Aleksander Weron
Part of the Progress in Probabilty book series (PRPR, volume 25)

Abstract

For a stationary process X we introduce the dynamical functional Φ by the formula Φ(Y,t)=Eexp(i(Y∘St-Y)), where Y belongs to the closure of the span {Xt:t∈ℝ} in the topology of convergence in measure and St is the shift transformation. It is proved that a stochastically continuous process X is ergodic if and only if for each Y we have
$$ \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int\limits_0^T \phi \left( {Y,t} \right)dt = {\left| {E{e^{iY}}} \right|^2}. $$

This characterization is applied to symmetric stable processes to reprove and unify two independent equivalent conditions for ergodicity presented in [2] and [7].

Keywords

Stable Process Stationary Gaussian Process Finite Dimensional Distribution Shift Transformation Symmetric Stable Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Krzysztof Podgórski
    • 1
  • Aleksander Weron
    • 1
  1. 1.Institute of MathematicsTechnical University of WroclawWroclawPoland

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