Product of sequence of uniform integrable random variables are uniformly integrable?

Question

If $\{X_i\}$ for $i \in I $ and $\{Y_j\}$ for $j \in J$ are uniformly integrable.Then prove that, $\{X_i+Y_j\}$ for $(i,j) \in I \times J$ is uniformly integrable.What about $ \{X_iY_j\} $ for $(i,j) \in I \times J$ ?

Answer 1

We have that for any $\epsilon>0$ there exist $K$ in $\mathbb{R}^+$ such that $$\mathbb{E}[|X_i|\,I_{|X_i|\geq K}]\leq\varepsilon,\qquad \mathbb{E}[|Y_j|\,I_{|Y_j|\geq K}]\leq\varepsilon $$ and since $|X_i+Y_j|\leq 2\max(|X_i|,|Y_j|)$, $\{X_i+Y_j\}_{(i,j)\in I\times J}$ is a uniformly integrable family.

However, $L^1$ is not a space that is closed under product, hence $\{X_i Y_j\}$ is not an uniformly integrable family in the general case. As an example, just take $X_i=Y_j=Z$ for any $(i,j)\in I\times J$, where $Z$ is a random variable having pdf: $$ f_Z(x) = \frac{3\sqrt{3}}{4\pi\,(1+|x|^3)} $$ We have $Z\in L^1\setminus L^2$, hence $\{X_i Y_j\}$ cannot be a uniformly integrable family.