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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!

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    $\begingroup$ $E[X_t]/t$ is not a statistic. $X_t/t$ might be one, but its expectation isn't. $\endgroup$
    – Glen_b
    Apr 2, 2013 at 9:06
  • $\begingroup$ If $S_1, ... S_n$ are (assumed to be) i.i.d., then $(X_t)$ is a renewal process. No ? $\endgroup$
    – Stéphane Laurent
    Apr 2, 2013 at 9:51
  • $\begingroup$ Ah, yes. So what I am really testing is whether $S_1, \dots, S_n$ are i.i.d. I guess? $\endgroup$
    – Dave White
    Apr 2, 2013 at 10:08

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