Conditional expectation of an $L^p$ random variable

Question

If $Y$, $X$ and $X_1,X_2,...$ are real random variables such that $|X_n|\leq|X|$ for every $n$ and both $X$ and $Y$ are in $L^p$ $(E[|X|^p] < \infty$ and $E[|Y|^p] < \infty)$ for some $p \geq 1$, is it true that $E[|X_n-Y|^p|\mathcal{F}] < \infty$ unniformly in $n$, i.e. $E[|X_n-Y|^p|\mathcal{F}] < A$ for every $n$ and an arbitrary conditioning sigma-field $\mathcal{F}$ ?

Answer 1

For any $p \geq 1$, we have

$$|x+y|^p \leq 2^p (|x|^p+|y|^p),$$

and therefore

$$\begin{align*} \mathbb{E}(|X_n-X|^p \mid \mathcal{F}) &\leq 2^p \mathbb{E}(|X_n|^p \mid \mathcal{F}) + 2^p \mathbb{E}(|Y|^p \mid \mathcal{F}) \\ &\leq 2^p \mathbb{E}(|X|^p \mid \mathcal{F}) + 2^p \mathbb{E}(|Y|^p \mid \mathcal{F}). \end{align*}$$

This shows that $\mathbb{E}(|X_n-Y|^p \mid \mathcal{F})$ is bounded uniformly if both $\mathbb{E}(|X|^p \mid \mathcal{F})$ and $\mathbb{E}(|Y|^p \mid \mathcal{F})$ are bounded (uniformly for arbitrary sigma-fields $\mathcal{F}$).

If these two random variables are not bounded, then we can, in general, not expect uniform boundedness of $\mathbb{E}(|X_n-Y|^p \mid \mathcal{F})$. Suppose e.g. that $\mathbb{E}(|X|^p \mid \mathcal{F})$ is not bounded. If we choose $Y := 0$ and $X_n := X$, then $$\mathbb{E}(|X_n-Y|^p \mid \mathcal{F}) = \mathbb{E}(|X|^p \mid \mathcal{F})$$ is also not bounded. A similar (counter)example can be constructed if $\mathbb{E}(|Y|^p \mid \mathcal{F})$ is not bounded.