Stopped random walk
Let $(X_n)_{n \in \mathbb{N}_0}$ be an unbiased random walk on $\mathbb{Z}$, starting at $X_0 = x > 0$, i.e., $X_n = X_0 + \sum_{k=1}^n Y_k$ for independent binary variables $Y_k$ assuming $\pm 1$ with equal probability. Further, let $\tau = \inf\{n; X_n = 0\}$ be the first step of hitting zero.
a) Decide whether the stopped process $X_n^\tau := X_{n \wedge \tau}$, $n \ge 0$, is
(i) a martingale, and
(ii) uniformly integrable.
b) Show
(i) There is $X_\infty \in L^1$ such that $(X_n^\tau) \xrightarrow{a.s.} X_\infty$,
(ii) but $(X_n^\tau)$ does not converge in $L^1$.
