Renewal processes

Let $\left(f_{n}: n \geq 1\right)$ be a probability distribution on the positive integers, and define a sequence $(u_{n}: n \geq 0)$ by $u_{0} = 1$ and $u_{n} = \sum_{r = 1}^{n} f_{r} u_{n-r}, n \geq 1$. Explain why such a sequence is called a renewal sequence, and show that $u$ is a renewal sequence if and only if there exists a Markov chain $U$ and a state $s$ such that $u_{n} = \mathbb {P}\left(U_{n} = s \mid U_{0} = s\right)$.

Let $\left\{ X_{i}: i \geq 1\right\}$ be the inter-event times of a discrete renewal process on the integers. Show that the excess lifetime $B_{n}$ constitutes a Markov chain. Write down the transition probabilities of the sequence $\left\{ B_{n}\right\}$ when reversed in equilibrium. Compare these with the transition probabilities of the chain $U$ of your solution to Exercise (8.3.1).

Let $(u_{n}: n \geq 1)$ satisfy $u_{0} = 1$ and $u_{n} = \sum_{r = 1}^{n} f_{r} u_{n-r}$ for $n \geq 1$, where $\left(f_{r}: r \geq 1\right)$ is a non-negative sequence. Show that:

Events occur at the times of a discrete-time renewal process $N$ (see Example (5.2.15)). Let $u_{n}$ be the probability of an event at time $n$, with generating function $U(s)$, and let $F(s)$ be the probability generating function of a typical inter-event time. Show that, if $\left|s\right| < 1$ :

Prove Theorem (8.3.5): Poisson processes are the only renewal processes that are Markov chains.