Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space.


Suppose $\{X\_n\}$ is a sequence of random variables satisfying :
$$
\sup\_{n}{\mathbb{E}(|X\_n|)} <\infty
$$
Suppose that
$$
\dfrac{M\_j}{2}<\int\_{j-1<|X\_{n}|\leq j}{|X\_{n}(t)|d\mathbb{P}(t)}\leq M\_j+\dfrac{1}{j^2} \qquad\forall n\geq 1 \text{ and }1\leq j\leq n^2
$$
with $M\_j>0$ such that $\sum\_{j=1}^{\infty}{M\_j}<\infty$.


Show that:
$$
 \{X\_n\mathbb{1}\_{X\_n\in[-n,n]}\}\text{ is uniformly integrable} 
$$


Any ideas, please?



$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable
Note that for any $K, n\in\mathbb{N}$ with $K<n$ we have that
$$\begin{align\*}
\mathbb{E}[|X\_n|1\_{X\_n\in[-n, n]}1\_{|X\_n|>K}] &= \mathbb{E}[|X\_n|1\_{n \geq |X\_n|>K}]\\
&=\int\_{n\geq |X\_n|>K} |X\_n|d\mathbb{P}\\
&= \sum\_{j=K+1}^n \int\_{j\geq |X\_n|>j-1} |X\_n|d\mathbb{P}\\
&\leq \sum\_{j=K+1}^n M\_j+1/j^2\\
&\leq \left(\sum\_{j=K+1}^\infty M\_j\right)+ \left(\sum\_{j=K+1}^\infty 1/j^2\right)
\end{align\*}
$$


Note that the final bound does not depend on $n$, so $\sup\_n \mathbb{E}[|X\_n|1\_{X\_n\in[-n, n]}1\_{|X\_n|>K}]\leq \left(\sum\_{j=K+1}^\infty M\_j\right)+ \left(\sum\_{j=K+1}^\infty 1/j^2\right)\overset{K\to\infty}{\longrightarrow} 0$. Hence, we have uniform integrability.


