Let $(X_n)$ be an iid. sequence of real, integrable random variables with $EX_1=a>0$. Let $S_n=X_1+...+X_n$, $n=1,2,...$ and $N_t:=\sup\{n\geq 1|S_1,...,S_n\leq t\}$, $t\geq 0$ where $\sup\emptyset=0$. Then the following holds:
- $P(N_t<\infty)=1\,\forall t\geq 0$
- $\lim_{t\to\infty} N_t=\infty$ $P$-almost surely
- $\lim_{t\to\infty} N_t/t=1/a$ $P$-almost surely
My textbook says this follows immediately from the strong law of large numbers, but I don't see this. In fact, I don't even know how I could prove this. Can anyone help me out here? Thanks.
$$\forall n\geqslant n_\varepsilon,\ a(1-\varepsilon)\leqslant\frac{S_n}n\leqslant a(1+\varepsilon)\implies\forall t\geqslant a(1+\varepsilon)n_\varepsilon,\ \frac1{a(1+\varepsilon)}\leqslant\frac{N_t}t\leqslant\frac1{a(1-\varepsilon)}$$