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
Exponential random variables
A simple case of Uniforms
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 \([0,1]\). What do you notice?
Consider \[F^d(y):=\frac{1}{\mu} \int_0 ^y [1-F(x)]dx,\] where \(F\) is the cdf for \(1+U_1\). Let \(Y\) have cdf \(F\). Now consider independent inter-arrival times, \(Y, X_2, X_3, \ldots\). Check, \(D_1=N(1) = N(0)\) and \(D_2' =N(2)- N(1)\); what do you observe?