I have i.i.d $\{\zeta_{n}\}$ defined as follow:
$$\{\zeta_{n}=\int_{\sigma_{n}}^{\tau_{n}}1_{(0,\infty)}(X(s))ds\}$$
where $X(s)$ is a one dimensional Brownian Motion and $\sigma_{1}=\min\{t:X(t)=0\}$, $\tau_{1}=\min\{t>\sigma_{1}:X(t)=-1\}$, $\sigma_{2}=\min\{t>\tau_{1}:X(t)=0\}$,$\tau_{2}=\min\{t>\sigma_{2}:X(t)=-1\}\cdots $ etc.
How to show that $\sum_{n} \zeta_{n}$ converges to infinity a.s.? I think to use strong law of large number, but I do not get satisfactory proof myself.