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)


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].


Stable Process Stationary Gaussian Process Finite Dimensional Distribution Shift Transformation Symmetric Stable Process 
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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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