# Renewal theory

#### Siron

##### Active member
Hi,

Let $N(t)$ be the renewal process based on the positive independent random variables $X_j, j \geq 0$, that is, $N(t)= \max\{n: \sum_{j=1}^{n} X_j \leq t\}$.

One can prove that $\limsup_{t} \mathbb{E}\left(\left[\frac{N(t)}{t}\right]^2\right)<\infty$. Now, prove that $\frac{N(t)}{t} \mapsto \frac{1}{\mu}$ implies almost surely that $\frac{\mathbb{E}[N(t)]}{t} \mapsto \frac{1}{\mu}$ where $u = \mathbb{E}[X_1]$.

Hint: First, prove that $\{\frac{N(t)}{t}: t>0\}$ is uniformly integrable.

Anyone? I tried to prove the hint but that didn't work. How can I use the hint to prove the claim?

Thanks!