uniformly integrability implies converges in probability

Question

A sequence $(X_n)$ is uniformly integrable if:\begin{equation}\limsup \mathbb{E}[|X_n|1_{\{|X_n|>b\}}] =0.\end{equation}

A sequence $(X_n)$ of random variables converges in probability towards the random variable $X$ if for all $\varepsilon > 0$:

\begin{equation} \lim_{n\to\infty}\Pr\big(|X_n-X| \geq \varepsilon\big) = 0 \end{equation}

Can we imply that uniformly integrability implies converges in probability? if not, can I have a counterexample?

Answer 1

No, uniform integrability does not imply convergence. Note that any sequence of bounded random variables is uniformly integrable, so a counterexample is given by $X_n = (-1)^n$.