Triangle Inequality for Conditional Expectation
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Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be an integrable random variable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let $\expect {\size X \mid \GG}$ be a version of the conditional expectation of $\size X$ given $\GG$.
Then we have:
- $\size {\expect {X \mid \GG} } \le \expect {\size X \mid \GG}$ almost everywhere.
Proof
From Conditional Expectation is Monotone, we have:
- $\expect {X^+ \mid \GG} \ge 0$ almost everywhere
and:
- $\expect {X^- \mid \GG} \ge 0$ almost everywhere
where $X^+$ and $X^-$ are the positive and negative parts respectively.
Now, almost everywhere we have:
\(\ds \size {\expect {X \mid \GG} }\) | \(=\) | \(\ds \size {\expect {X^+ - X^- \mid \GG} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \size {\expect {X^+ \mid \GG} - \expect {X^- \mid \GG} }\) | Conditional Expectation is Linear | |||||||||||
\(\ds \) | \(\le\) | \(\ds \size {\expect {X^+ \mid \GG} } + \size {\expect {X^- \mid \GG} }\) | Triangle Inequality for Real Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X^+ \mid \GG} + \expect {X^- \mid \GG}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X^+ + X^- \mid \GG}\) | Conditional Expectation is Linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {\size X \mid \GG}\) | Sum of Positive and Negative Parts |
$\blacksquare$