Incidents for product $A$ occur at time $T_1, T_2,\dots$ where $T_i=X_1+X_2+\dots+X_i$. Assume that $(X_i)_i$ are i.i.d. and let $M(t)=\max\{n:T_n \leq t\}$ the number of incidents occured at time $t$. Show that $$\frac{M(t)}{t}\to\frac{1}{E[X_1]}$$ a.e.
I was thinking to apply the SLLN. Denote $\bar{G_t}=\frac{1}{t} \sum_{i=1}^t m(i)$. From the SLLN we know $$\mathbb{P}\left(\lim_{t\to\infty} \bar{G_t}=\Bbb E(\bar{G_t})\right)=1 (iid)$$ I am however stuck at this point. Could anyone give me a hint?