Suppose $\{X_n\}$ and $\{Y_n\}$ are two families of u.i(uniform integrable) random variables defined on the same probability space. Is $\{X_n+Y_n\}$ u.i?
Proof
Given $$\mathbb{E}[|X_n|\,I_{|X_n|\geq K}]\leq \frac{\varepsilon}{4} ,\qquad \mathbb{E}[|Y_n|\,I_{|Y_n|\geq K}]\leq \frac{\varepsilon}{4} $$ Since $|X_n+Y_n|\,I_{|X_n+Y_n|\geq 2K}\leq 2|Y_n|\,I_{|Y_n|\geq K}+2|X_n|\,I_{|X_n|\geq K}$ Then $$\mathbb{E}[|X_n+Y_n|\,I_{|X_n+Y_n|\geq 2K}]\leq\mathbb{E}[2|Y_n|\,I_{|Y_n|\geq K}+2|X_n|\,I_{|X_n|\geq K}]$$$$=2\mathbb{E}[|Y_n|\,I_{|Y_n|\geq K}]+2\mathbb{E}[|X_n|\,I_{|X_n|\geq K}]\leq 2 \frac{\varepsilon} {4}+2 \frac{\varepsilon} {4}= \varepsilon$$ Thus $\{X_n+Y_n\}$ is ui.