(a) Define the Moran model and Kingman's $n$-coalescent. Define Kingman's infinite coalescent.

Show that Kingman's infinite coalescent comes down from infinity. In other words, with probability one, the number of blocks of $\Pi_{t}$ is finite at any time $t>0$.

(b) Give the definition of a renewal process.

Let $\left(X_{i}\right)$ denote the sequence of inter-arrival times of the renewal process $N$. Suppose that $\mathbb{E}\left[X_{1}\right]>0$.

Prove that $\mathbb{P}(N(t) \rightarrow \infty$ as $t \rightarrow \infty)=1$.

Prove that $\mathbb{E}\left[e^{\theta N(t)}\right]<\infty$ for some strictly positive $\theta$.

[Hint: Consider the renewal process with inter-arrival times $X_{k}^{\prime}=\varepsilon \mathbf{1}\left(X_{k} \geqslant \varepsilon\right)$ for some suitable $\varepsilon>0$.]