Playing with renewal processes
In this exercise, we will demonstrate what happens when we look at a
renewal process \(N(t)\) for a large
value of \(t \approx 150\). Consider
the following simple checks:
- Estimate by simulations the distribution of \(D_1=N(1) - N(0)\), \(D_2=N(151) - N(150)\), and \(D_3=N(161) - N(160)\). If \(N\) is stationary, then \(D_1, D_2, D_3\) should all have the same
distribution
- Run this check on a Poisson process, generated by exponential
inter-arrival times
- Run this check on a the renewal process with inter-arrival times
where \(X_i = 1+ U_i\), where \(U_i\) are uniformly distributed on \([0,1]\).