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On time- and cycle-stationarity

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  • Thorisson, Hermann

Abstract

Consider processes split into cycles by a sequence of random times (called points). The standard Palm relationship between stationary processes with cycles and processes with stationary cycles is produced in two transparent steps: length-biasing and re-centring. It has the following standard intuitive interpretation: the process with stationary cycles behaves like the stationary one conditioned on a point at time zero. A less known modification of this relationship is produced by conditioning on the invariant [sigma]-algebra before length-biasing. It has the following intuitive interpretation: the stationary process behaves like the cycle-stationary one centred at a time chosen at random on the line. The present approach leads to strong conditioning, limit and coupling results motivating these interpretations.

Suggested Citation

  • Thorisson, Hermann, 1995. "On time- and cycle-stationarity," Stochastic Processes and their Applications, Elsevier, vol. 55(2), pages 183-209, February.
    • Handle: RePEc:eee:spapps:v:55:y:1995:i:2:p:183-209
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    References listed on IDEAS

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    1. Thorisson, Hermann, 1992. "Construction of a stationary regenerative process," Stochastic Processes and their Applications, Elsevier, vol. 42(2), pages 237-253, September.
    2. Glynn, Peter & Sigman, Karl, 1992. "Uniform Cesaro limit theorems for synchronous processes with applications to queues," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 29-43, February.
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    Cited by:

    1. Nieuwenhuis, G., 1996. "Ergodicity Conditions and Cesaro Limit Results for Marked Point Processes," Other publications TiSEM e9963ce3-420e-49b5-88cc-9, Tilburg University, School of Economics and Management.
    2. Nieuwenhuis, G., 1996. "Ergodicity Conditions and Cesaro Limit Results for Marked Point Processes," Research Memorandum 736, Tilburg University, School of Economics and Management.
    3. Christian Mönch, 2022. "Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1842-1862, September.
    4. Last, Günter, 1996. "Coupling with compensators," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 147-170, December.
    5. Bardhan, Indrajit, 1995. "Further applications of a general rate conservation law," Stochastic Processes and their Applications, Elsevier, vol. 60(1), pages 113-130, November.

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