Renewal theory

Remarks

The condition $P(T_1 = 0) < 1$ makes $S_n \to \infty$ almost surely, so $N(t)$ is finite for each $t$. Example: if the $T_i$ are exponential with rate $\lambda$, then $N$ is the Poisson process of rate $\lambda$, and $m(t) = \lambda t$. Non-example: if $T_1, T_2, \ldots$ are not identically distributed, for instance $T_i = 2^{-i}$, then $N$ is not a renewal process, and here $N(t) = \infty$ for $t \ge 1$. Let $\mu = \mathbb{E}[T_1] \in (0, \infty]$ be the expected value of a holding time. The elementary renewal theorem says $m(t)/t \to 1/\mu$ as $t \to \infty$, with $1/\infty = 0$; the strong law gives $N(t)/t \to 1/\mu$ almost surely.