# probability – Going “well into the lifetime” of a renewal process means the time until the next event will be uniform conditional on inter-arrival?

Blackwell’s theorem (theorem 4.6.2 here) on renewal processes says (for non-arithmetic renewal processes) that if $$m(t)$$ is the average number of events occurring from the start of the process to time, $$t$$ and $$delta$$ is any positive real number,

$$lim_{tto infty} (m(t+delta)-m(t)) = frac{delta}{E(X)}$$

Here, $$E(X)$$ is the mean time between events for the process. In words, if you go “well into the lifetime” of a renewal process and count the number of events in an interval of length $$delta$$, you will count on average $$frac{delta}{E(X)}$$ arrivals.

I conjecture that what going “well into the lifetime” is doing is ensuring that the conditional distribution of the time from the start of your observation window to the first event seen after it is uniform over $$(0,t)$$ conditional on $$t$$ being the inter-arrival time for the interval within which the start of our observation window landed. I’m looking for a proof of this conjecture or a counter-example to show it isn’t true.

Why I’m making this conjecture

Consider a deterministic point process (with time between arrivals deterministic, $$t$$). This is a kind of renewal process. For this special case, Blackwell’s theorem can be proved as case-2 in the question here: General point process – expected number of arrivals within an interval.

Further, it’s shown here: For what distributions, \$X\$ is \$E(lfloor c-X rfloor) = c-1\$? that the proof only works when the time from the start of the window to the next event is uniform over $$(0,t)$$.

If this condition of uniformity is required for the deterministic point process, it must be required in general as well.