I have to prove something about normaly distributed variables.
Let $X_1, X_2, ... $ be i.i.d. with normal distribution $N(\mu, \sigma^2)$, where $\mu >0$. Define: $S_n := X_1 + X_2 + ... + X_n$ and $Z_n := \max\{S_0, S_1, S_2, ..., S_n\}$, where $S_0=0$.
I must show that random variable $Z_{\infty} = \max\{S_0, S_1, S_2, ...\}$ is almost surely finite, which means that $P(Z_{\infty} < \infty)=1$.
Any help will be very appreciated. Thanks.