Uniformly integrable random variables remain uniformly integrable after centering

Question

Let $\{X_i\}_{i \in I}$ be a familiy of uniformly integrable random variables. I want to show that the family $\{X_i - E[X_i]\}_{i \in I}$ has the same property, i.e. $$ \forall \varepsilon > 0 \exists C > 0: \sup_{i \in I} E\big[\mathbf{1}_{\{ |X_i - E(X_i) | > C\}} \, |X_i - E[X_i]| \big] \leq \varepsilon. $$ How can I show this?

Answer 1

$\newcommand{\U}{\mathscr{U}}\newcommand{\P}{\mathbb{P}}\newcommand{\E}{\mathbb{E}}$Let $\U$ be a uniformly integrable family of random variables over a probability space $(\Omega,\mathfrak{M},\P)$. Now define $$\U_c=\{X-\E(X)~:~X\in\U\}$$ to be the set of centered random variables. We claim that $\U$ is bounded in $L^1(\Omega,\P)$. Fix any $\varepsilon>0$ and let $C$ be the corresponding constant as in your definition of uniform integrability. Now for any $X\in\U$, $$\E|X|\leq E(|X|;|X|\leq C)+\E(|X|;|X|>C)\leq C\P(|X|\leq C)+\varepsilon\leq C + \varepsilon.$$ So we see that $\U$ is bounded in $L^1$, and in turn we deduce by Minkowski's inequality that $\U_c$ is bounded in $L^1(\Omega,\P)$. Let $M$ denote the constant which uniformly bounds $\U_c$ in $L^1$.

Let $\varepsilon$ and $C$ be as above. Now fix some $X\in\U$, and we will consider the corresponding random variable $X_c=X-\E(X)\in\U_c.$ By Markov's inequality we see that $\P(|X-\E(X)|>K)\leq M/K$ for any any $K$, and so \begin{align*} \E(|\E(X)|;|X-\E(X)|>K)&=\E(X)|\P(|X-\E(X)|>K)\leq \frac{M^2}{K} \end{align*} We also have the following estimate \begin{align*} \E(|X|;|X-\E(X)|>K)&\leq \E(|X|;|X-\E(X)|>K,|X|\leq C) + \E(|X|;|X-\E(X)|>K,|X|>C) \\ & \leq C\P(|X-\E(X)|>K)+\varepsilon\\ & \leq \frac{MC}{K}+\varepsilon. \end{align*} Now by taking a sufficiently large $K$ we can bound both of the above estimates uniformly by $2\varepsilon$. Now by Minkowski's inequality again we have \begin{align*} \E(|X-\E(X)|;|X-\E(X)|>K)&\leq\E(|X|;|X-\E(X)|\geq K)+\E(|\E(X)|;|X-\E(X)|>K) \\ &\leq 4\varepsilon. \end{align*} Since $\varepsilon$ was arbitrary and since $K=K(M,C,\varepsilon)$, we see that $\U_c$ is a uniformly integrable family.

Note: There is an equivalent characterization/definition of uniform integrability from which this result follows more or less immediately from. A family of random variables $\U$ is unifomly integrable if and only if $\U$ is bounded in $L^1$ and for all $\varepsilon>0$ there exists some $\delta >0$ such that for all measurable $F$ with $\P(F)<\delta$ and $X\in\U$, then $\E(|X|;F)<\varepsilon$.