That Brownian Motion's increments are gaussian is "not surprising"?

Question

In section 1 of chapter 1 of Continuous Martingales and Brownian Motion, the authors claim that

the fact that the increments of of Brownian motion are gaussian random variables "is not surprising in view of the central-limit theorem".

Why do they put a hyphen between 'central' and 'limit'? Just kidding, my real question is: What do they mean by this?

My understanding of the CLT is limited to, "the averages of same-size samples from any distribution converges in distribution to a gaussian distribution" (and please correct me if that's wrong), but I don't see how that applies here.

Answer 1

"the averages of same-size samples from any distribution converges in distribution to a gaussian distribution" (...) don't see how that applies here.

It applies if you treat the increment from $B(0)$ to $B(t)$ as a sum of $n$ independent identically distributed increments $B(t_{i})-B(t_{i-1})$, where $t_i = \frac{i}{n} t$. Just divide the time interval into $n$ equal pieces and take the limit of large $n$.

Just as non-surprisingly, the same argument applies to time-changed Brownian motion $B(f(t))$. The subintervals should be equal in variance of the increments, instead of time.

Answer 2

The context is the "real" Brownian Motion, the one with the pollen and water particules.

A "2D-increment" in the trayectory of the pollen is the result of a very large number of identically distributed and independent shocks. Then, by the CLT (under finite variance hypothesis) the sum of the shocks will be asymptotically normal.