Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 2 / 20H / a

Let $S_k={\sum }_{\ell =1}^k{Y}_{\ell }$, $S_0=0$, be the epochs of the renewal process. The definition uses the next epoch at or after $n$, so $X_n=0$ at an epoch. If $X_n=i\ge 1$, no renewal occurs before $n+i$ and necessarily $X_{n+1}=i-1$.

If $X_n=0$, time $n$ is a renewal epoch, and the next unused interarrival time is independent of the entire observed past with the original distribution. To justify the random index, condition on each possible number $k$ of renewals up to $n$: the event identifying $k$ depends only on $Y_1,\dots ,Y_k$, whereas $Y_{k+1}$ is independent of those variables. Summing over $k$ preserves that distribution and independence. Therefore the residual lifetime Markov chain has transition probabilitiesIts reachable state space is $E=\{j\ge 0:\mathbb{P}(Y_1>j)>0\}$, either an initial finite segment or all nonnegative integers. The conditional next-step law depends only on $X_n$, proving the Markov property. Initially $X_0=0$. The reset is $Y-1$, rather than $Y$, because one unit of time has elapsed by the next step.