Let $(Z_n)_{n\ge0}$ be a sequence of i.i.d. r.v.'s with $P(Z_1=1)=P(Z_1=-1)=\frac{1}{2}$. Let $S_0=0$ and $S_n=Z_1+\dots+Z_n$. Let $\mathcal{F}_0=\{\emptyset,\Omega\}$ and $\mathcal{F}_n=\sigma(Z_1,\dots,Z_n)$. Let $a>0$, $0<\lambda<\frac{\pi}{2a}$, and $T=\inf\{n\ge1:|S_n|=a\}$.
So far, I have shown that $$ X_n=(\cos{\lambda})^{-n}\cos{(\lambda S_n)} $$ is a martingale and that $T$ is almost surely finite. Now, I must further prove that $(X_{n\wedge T})_{n\ge0}$ is uniformly integrable, but I don't see how.
Could someone give me a hint?