Consider a standard Brownian motion $B_t$ with $B_0 = 0$.
I would like to show that if $\pi_n$ is a refining sequence of partitions of $[0,\infty)$ whose mesh tends to $0$, then $B_t^n = \sum_{t_k \in \pi_n} B_{t_k} 1_{(t_k, t_{k+1}]}$ converges to $B_t$ in ucp, i.e.
for all $t \ge 0$, $$ \lim_{n\to \infty}P(\sup_{0 \le s \le t}|B_s^n – B_s| >\epsilon) \to 0$$ for all $\epsilon>0$.
I know that on compact intervals, the Brownian motion is uniformly continuous, so we can choose the mesh small enough that for each fixed $\omega$, we can take the inner event to be empty. I.e. we could get $P(\lim_n \sup_{0 \le s \le t} |B_s^n – B_s|>\epsilon)=0$. However, I cannot figure out how to take this limit outside of the probability.
How can I prove this result?
Best Answer
All you need is $\lim \sup P(E_n) \leq P( \lim \sup E_n)$ which follows by applying Fatou's Lemma to then sequence $(I_{E_n^{c}})$.