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The superposition of the backward and forward processes of a renewal process

Author

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  • Coleman, Rodney

Abstract

A fixed sampling point O is chosen independently of a renewal process on the whole real line. The distances Y1, Y2, ... from O to the renewal points of , when they are measured either forwards or backwards in time, define a point process . The process is a folding over of the past of onto its future. It is the superposition of two equilibrium renewal processes which are known to be independent only when is a Poisson process. The joint distribution of Y1, Y2, ..., Yk is found. The marginal distribution of 2Yk is shown to be the same as that of the distance from O to the kth following point of . The intervals of are shown to have a stationarity property, and it is proved that if any pair of adjacent intervals of are independent, then is a Poisson process.

Suggested Citation

  • Coleman, Rodney, 1976. "The superposition of the backward and forward processes of a renewal process," Stochastic Processes and their Applications, Elsevier, vol. 4(2), pages 135-148, April.
    • Handle: RePEc:eee:spapps:v:4:y:1976:i:2:p:135-148
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