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

AbstractA 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