Is the product of two independent uniform integrable random variable is uniform integrable?

Question

Is the product of two independent uniform integrable random variable is uniform integrable? What is the role independence plays here?

Answer 1

If $(X_i)_{i\in I}$ and $(Y_i)_{i\in I}$ are two families of random variables such that

• for each $i\in I$, $X_i$ and $Y_i$ are independent;
• the family $(X_i)_{i\in I}$ is uniformly integrable;
• the family $(Y_i)_{i\in I}$ is uniformly integrable,

then the family $(X_iY_i)$ is uniformly integrable.

To see that, notice that $\{|X_iY_i|\gt R\}\subset \{|X_i|\gt \sqrt R\}\cup\{|Y_i|\gt \sqrt R\}$, hence $$\int_{\{|X_iY_i|\gt R\}}|X_iY_i|\mathrm d\mu\leqslant \int_{\{|X_i|\gt \sqrt R\}}|X_i||Y_i|\mathrm d\mu+\int_{\{|Y_i|\gt \sqrt R\}}|X_i||Y_i|\mathrm d\mu.$$ Using now independence, we obtain $$\int_{\{|X_i|\gt \sqrt R\}}|X_i||Y_i|\mathrm d\mu=\mathbb E|Y_i|\cdot \int_{\{|X_i|\gt \sqrt R\}}|X_i|\mathrm d\mu\leqslant \sup_{j\in I}\mathbb E|Y_j|\cdot \int_{\{|X_i|\gt \sqrt R\}}|X_i|\mathrm d\mu$$ and we conclude since $\sup_{j\in I}\mathbb E|Y_j|$ is finite.