The renewal equation

Prove that $\mathbb {E}\left[e^{\theta N(t)}\right] < \infty$ for some strictly positive $\theta$ whenever $\mathbb {E}\left[X_{1}\right] > 0$. [Hint: Consider the renewal process with interarrival times $X_{k}^{\prime } = \epsilon I_{\left\{ X_{k} \geq \epsilon \right\} }$ for some suitable $\epsilon$.]

Let $N$ be a renewal process and let $W$ be the waiting time until the length of some interarrival time has exceeded $s$. That is, $W = \inf \left\{ t: C(t) > s\right\}$, where $C(t)$ is the time which has elapsed (at time $t$) since the last arrival. Show that

where $F$ is the distribution function of an interarrival time. If $N$ is a Poisson process with intensity $\lambda$, show that

and $\mathbb {E}\left[W\right] = \left(e^{\lambda s}-1\right) / \lambda$. You may find it useful to rewrite the above integral equation in the form of a renewal-type equation.

Find an expression for the mass function of $N(t)$ in a renewal process whose interarrival times are:

Let the times between the events of a renewal process $N$ be uniformly distributed on $(0,1)$. Find the mean and variance of $N(t)$ for $0 \leq t \leq 1$.

Let $N$ be the renewal process with interarrival times $X_{1}, X_{2}, \ldots$ Show that, for $t > 0$, the interarrival time $X_{N(t)+1}$ is stochastically larger than $X_{1}$.

Suppose the interarrival times $X_{i}$ of a renewal process have a density function $f$ with ordinary Laplace transform $\widehat{f}(\theta ) = \int_{0}^{\infty } e^{-\theta x} f(x) d x$. Show that the renewal function $m$ has Laplace transform

Let $r(y)$ be the ruin function of the insurance problem (8.10.7). In the notation of that problem, show that the Laplace-Stieltjes transforms $r^{*}(\theta )$ and $F^{*}(\theta )$ are related by $\lambda F^{*} = r^{*}\left(\theta -\lambda +\lambda F^{*}\right)$.