Uniform integrability of random variables $~e^{(X_1+X_2+...+X_{n-1}+X_n - \frac{n}{2})}$

Question

I have a task: $(X_n)$ are i.i.d. $$P(X_n=1)=P(X_n=-1)=\frac{1}{2}~.$$ Prove that $$Z_n=e^{(X_1+X_2+...+X_{n-1}+X_n - \frac{n}{2})}$$ is uniformly integrable.

We have to prove that $$\lim_{b \rightarrow \infty} \sup_{n} EZ_n\mathbf{1}_{Z_n>b}=0~,$$

but $Z_n$ is not limited. How prove this? Thanks in advance.

Answer 1

Hint: we can compute explicitely $\mathbb E\left[Z_n^p\right]$ for $p>1$. We will find something like $a_p^n$. Then choose $p>1$ such that $a_p<1$.