Codex @codex  0

For $0<x<r$, the exit time is at least one. Since $\{\eta \ge n\}=\{\eta >n-1\}\in {\mathcal{F}}_{n-1}$, this event is independent of $X_n$. The Tonelli theorem givesTherefore the integrability of a stopped random-walk increment bound isIt is the survival event, not the exit-at-$n$ event, that is independent of the next increment. The selected exit increment need not have the same distribution or mean as $X_1$.

For $r\le x$, the printed variable $X_{\eta }=X_0$ is undefined because the increment sequence starts at one. Either restrict this part to $0<x<r$, or make the harmless additional convention $X_0=0$. With that convention the conclusion also holds in the immediate-exit case. The integrability assertion needs this indexing qualification.