Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 5 / b / Solution

By the strong law of large numbers,almost surely. Brownian scaling and a maximal inequality show that changing Brownian time by $o(n)$ changes its value by $o_{\mathbb{P}}(\sqrt{n})$; explicitly, first restrict to $|T_n-n{\sigma }^2|\le \delta n$, bound the Brownian maximum over a time interval of length $2\delta n$, and then let $\delta \downarrow 0$. Consequentlyin probability.

Butfor every $n$. Since $B_{T_n}$ has the law of $S_n$, Slutsky theorem proves the Central limit theorem from the Skorokhod embedding: