Abstract
We prove first that a renewal process is stationary if and only if the distributions of the age and the residual waiting time coincide for every t>0, and for 0≦x<t. From there it follows a characterization of memoryless distributions (i.e., either exponential or geometric,) in the case of an ordinary renewal process.
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Garagorry, F.L., Ahsanullah, M. A characterization of stationary renewal processes and of memoryless distributions. Statistische Hefte 18, 46–48 (1977). https://doi.org/10.1007/BF02932905
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DOI: https://doi.org/10.1007/BF02932905