$T$ can be $\infty$ with positive probability From Williams' Probability with Martingales





How exactly do we know $T$ can be $\infty$ with positive probability
or
$$P(T = \infty) > 0 \text{ ?}$$
I'm guessing that that means there is a positive probability that for some $c$, the sum, $X_1 + \cdots + X_r$, will never exceed $c$ no matter how large $r$ is.
If the partial sums are bounded, then no matter how many terms in the sequence we add, they won't exceed some $c$ so
$$\{r \mid |X_1 + X_2 + \cdots + X_r| > c\} = \emptyset \text{ ?}$$
Why can't we say that $P(T = \infty) = 1$?

Indicating the dependence of $T$ on $c$ explicitly, you have
$$\{T_c=\infty\}=\{\sup_n|M_n|\le c\}$$
Therefore,
$$\bigcup_{c>0,c\in\Bbb Q}\{T_c=\infty\}=\{\sup_n|M_n|<\infty\}$$
