Let $B$ be standard Brownian motion started at 0, and define the Brownian bridge on $[0,1]$ by

\[\begin{aligned} \widehat{B}_t &= B_t - t B_1 \qquad 0 \le t \le 1 . \end{aligned}\]

a. Show that the Brownian bridge is a continuous Gaussian process (i.e., that finite dimensional distributions are all Normal) with

\[\begin{aligned} \mathbb{E}[\widehat{B}_t] &= 0 \\ \mathbb{E}[\widehat{B}_s \widehat{B}_t] &= s (1-t) \quad \text{for } 0 \le s \le t \le 1 . \end{aligned}\]

b. Show that if we let $( \widehat{B}^\epsilon_t )_{t\in[0,1]}$ be the process $B$ conditioned on ${ |B_1| < \epsilon }$, then $\widehat{B}^\epsilon$ converges in distribution to $\widehat{B}$.

Attribution: This problem is modified from worksheets provided by Jim Pitman (thanks!).