I have $n$ realisations $s_1,\, \dots , s_n$ of random variables $S_1,\, \dots, S_n$ which are assumed to be i.i.d. with unknown distribution. These measure the time between events.
I want to calculate the probability that the data can be modelled by a renewal process: $(X_t)_{t\geq0}$ where $X_t$ is the total number of jumps by time $t$.
In particular, I am using the elementary renewal equation $\lim_{t\rightarrow \infty} \frac{E[X_t]}{t} = \frac{1}{E[S_i]}$ and I want the probability that the observed values of the LHS and RHS agree for a given $(n, t)$.
Do I want to some how construct a confidence interval for the statistic $\frac{E[X_t]}{t}$ and test whether $\frac{1}{E[S_i]}$ lies in this interval?
I guess what I'm asking, is how can I calculate the error in the strong law of large numbers for a given $n$, I think?
Apologies for the confusing description, my statistics isn't the best!
