Consider a simple random walk $(X_n)_{n\in\Bbb N}$ on $\Bbb Z$ with transition probabilities $p_{i,i+1}=p=1-p_{i,i-1}$ for all $i$ and $1/2<p<1$. In this question (What are some martingales for asymmetric random walks?), it is shown how martingales involving powers of $X_n$ may be obtained.
Consider as an example the martingale $Y_n:=(X_n-(2p-1)n)^2-4p(1-p)n$. Suppose $X_0=0$ and let $\tau$ be the first hitting time of $1$.
Question How can we show that the stopped martingale $Y_{\tau\wedge n}$ is uniformly integrable (UI)?
More generally, I would like to show that stopped martingales involving higher powers of $X_n$ are also UI. The motivation for this is that we can then use the optional sampling theorem to compute the moments of $\tau$.
From general Markov chain theory $\Bbb E[\tau]<\infty$, so the only difficulty is showing that the first term of $Y_{\tau\wedge n}$, viz. $(X_{\tau\wedge n}-(2p-1)(\tau\wedge n))^2$ is UI. I am not sure how to do this.