Today we had another lecture about renewal processes. The focus was on variability of count statistics.
We have seen a theorem relating the asymptotics of the Fano factor, which is the normalized variance of the counts, and the coefficient of variation of the inter event times.
If you have a point process, you can think of two different types of variability:
1) variability within each realization: how inter event times do differ from each other;
2) variability across trials: how statistics change from one realization to the next.
It is quite intuitive that for renewal processes both type of variability should be somehow connected: if you have a long realization of a renewal process, you can cut that into pieces and construct many shorter realizations; the renewal property implies that the statistics of the every single realization will be independent from the others.
This is exactly the meaning of the theorem about the asymptotics of the Fano factor: in the limit, for a renewal process the Fano factor and the CV2 will coincide.