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Lévy's upward theorem:

Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} \mathcal{F}_n)$.

Define $X_n = E[Y | \mathcal{F}_n]$ and $Z = E[Y | \mathcal{F}_{\infty}]$.

Then $X_n \to Z$ a.s. (and in $\mathcal{L}^1$)

Now for my problem suppose that $p > 1$ and $Y \in \mathcal{L}^p(\Omega, \mathcal{F}, P)$.
How to show that $X_n \to Z$ in $\mathcal{L}^p$ ?

I thought about applying the dominated convergence theorem but I could not find a right bound for $|X_n - Z|^p$.

Any ideas? Thanks.

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  • $\begingroup$ which book is it ? $\endgroup$ – WLOG Jan 28 '15 at 13:23
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We have to show that if $ X\in\mathbb L^p$ and $X_n:=\mathbb E[X\mid \mathcal F_n]$, then $\mathbb E|X_n-\mathbb E[X\mid \mathcal F_\infty|^p\to 0$.

We can use a truncation argument: for a fixed $R$ we have $$\mathbb E|X_n-Y|^p\leqslant 2^{p-1}\mathbb E\left|\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_\infty]\right|^p+\\ +2^{p-1}\mathbb E\left|\mathbb E[X\chi\{|X|\gt R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\gt R\}\mid\mathcal F_\infty]\right|^p.$$ Since $|\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_\infty]|$ is bounded by $2R$, we get $$\mathbb E|X_n-Y|^p\leqslant 2^{p-1}(2R)^{p-1}\mathbb E\left|\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_n]-\mathbb E[X\chi\{|X|\leqslant R\}\mid\mathcal F_\infty]\right|\\ +4^{p-1}\mathbb E|X\chi\{|X|\gt R\}|^p.$$ Using the $\mathbb L^1$ case with $Y:=X \chi\{|X|\leqslant R\}$, we obtain $$\limsup_{n\to \infty}\mathbb E|X_n-Y|^p\leqslant 4^{p-1}\mathbb E|X\chi\{|X|\gt R\}|^p.$$ Since this bound is true for any $R$, we obtain the wanted conclusion by monotone convergence.

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  • $\begingroup$ I'm sorry, what's X in $X_{\mathcal{X}} \{|X| \leq R \}$? $\endgroup$ – MrReese Nov 24 '14 at 17:53
  • $\begingroup$ I'v edited in order to clarify the notations. $\endgroup$ – Davide Giraudo Nov 24 '14 at 18:07
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Hints:

  1. Use Jensen's inequality in order to prove that $$\mathbb{E}(|X_n|^p) \leq \mathbb{E}(|Y|^p).$$
  2. Deduce from $$\sup_{n \geq 1} \mathbb{E}(|X_n|^p)< \infty$$ and Doob's maximal inequality that $$\sup_{n \geq 1} |X_n| \in L^p.$$
  3. Apply the dominated convergence theorem.
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  • $\begingroup$ Thanks. I accepted the other answer because Doob's maximal inequality appears later in the book. $\endgroup$ – MrReese Nov 24 '14 at 19:04

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