I want to determine the distribution $\rho(t_{wait})$ given the distribution of time gaps between random events, not necessarily Poisson. Here $t_{wait}\equiv$ time a random arrival spends waiting until the next event.
In the case of a Poisson process, $$\rho(t_{gap}) = \rho(t_{wait}) = \lambda\exp(- \lambda t)\tag{1}$$
because the process has no memory; it doesn't matter if an event just occurred, the times $t_{gap}$ are the same as $t_{wait}$.
However, for a general process, if I know $\rho(t_{gap})$, can I say anything about $\rho(t_{wait})$? It's mean, for instance? If not, what extra information do I need?
EDIT: To be clear, the "random arrival" is drawn from a uniform distribution between events.