Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 21 / 4 / c / Solution

True. The unit-time increments $Z_k=B_k-B_{k-1}$ are independent and identically distributed random variables with standard normal distribution, hence integrable with mean zero. The strong law of large numbers gives $B_n/n\to 0$ almost surely at integer times.

To control the intervals between integers, put $M_n={\sup }_{0\le u\le 1}|B_{n+u}-B_n|$. Stationary increments and the Doob L2 maximal inequality give $\mathbb{E}{M}_n^2\le 4$. Thus the Markov inequality yieldsThe Borel-Cantelli lemma, followed by a countable intersection over positive rational $\epsilon$, implies $M_n/n\to 0$ almost surely. For $n\le t\le n+1$,Both terms tend to zero on the same probability-one event, proving $\boxed{B_t/t\to 0\text{ almost surely}}$ for continuous time as well.