I need to construct a counter example such that the process $\{X_n\}_{n \ge 1}$ is uniformly integrable; however, the stopped process $X_{\tau \wedge n}$ where $\tau$ is a stopping time, is NOT uniformly integrable. I thought it will be easier to consider these two cases: 1. $X_n$ be a martingale; 2. $\mathbb{E}[\tau] = \infty$; However the example does not need to meet neither of 1 and 2. Any ideas?
2026-03-26 16:03:46.1774541026
Stopped process not uniformly integrable
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