Conditional Reverse Fatou's Lemma

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be an integrable random variable.

Let $\sequence {X_n}_{n \mathop \in \N}$ be an sequence of non-negative integrable random variables such that:

there exists an integrable random variable $Y$ such that:

$\size {X_n} \le Y$ almost surely.

Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.

For each $n \in \N$, let $\expect {X_n \mid \GG}$ be a version of the conditional expectation of $X_n$ conditioned on $\GG$.

Let $\expect {X_n \mid \GG}$ be a version of the conditional expectation of $X_n$ conditioned on $\GG$.

Let $\ds \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}$ be a version of the conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$.

Then:

$\ds \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG} \le \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}$ almost surely.

Proof

We should first verify that a version of the conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$ exists.

We have:

$-Y \le X_n \le Y$

and so:

$\ds -Y \le \limsup_{n \mathop \to \infty} X_n \le Y$ almost surely

so that:

$\ds -\infty < -\expect Y \le \expect {\limsup_{n \mathop \to \infty} X_n} \le \expect Y < \infty$

from Expectation is Monotone.

So:

$\ds \limsup_{n \mathop \to \infty} X_n$ is integrable

and we have that conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$ exists.

Now, we have that:

$Y - X_n \ge 0$

So, we can apply Conditional Fatou's Lemma to obtain:

$\ds \expect {\liminf_{n \mathop \to \infty} \paren {Y - X_n} \mid \GG} \le \liminf_{n \mathop \to \infty} \expect {Y - X_n \mid \GG}$ almost surely.

We have, by Conditional Expectation is Linear:

$\ds \expect {Y - X_n \mid \GG} = \expect {Y \mid \GG} - \expect {X_n \mid \GG}$

So we have:

\(\ds \liminf_{n \mathop \to \infty} \expect {Y - X_n \mid \GG}\)

\(=\)

\(\ds \liminf_{n \mathop \to \infty} \paren {\expect {Y \mid \GG} - \expect {X_n \mid \GG} }\)

\(\ds \)

\(=\)

\(\ds \expect {Y \mid \GG} + \liminf_{n \mathop \to \infty} \paren {-\expect {X_n \mid \GG} }\)

Infimum Plus Constant

\(\ds \)

\(=\)

\(\ds \expect {Y \mid \GG} - \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG}\)

Negative of Supremum is Infimum of Negatives

Similarly:

\(\ds \expect {\liminf_{n \mathop \to \infty} \paren {Y - X_n} \mid \GG}\)

\(=\)

\(\ds \expect {Y + \liminf_{n \mathop \to \infty} \paren {-X_n} \mid \GG}\)

Infimum Plus Constant

\(\ds \)

\(=\)

\(\ds \expect {Y - \limsup_{n \mathop \to \infty} X_n \mid \GG}\)

Negative of Supremum is Infimum of Negatives

\(\ds \)

\(=\)

\(\ds \expect {Y \mid \GG} - \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}\)

Conditional Expectation is Linear

So we have:

$\ds \expect {Y \mid \GG} - \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG} \le \expect {Y \mid \GG} - \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG}$ almost surely.

So:

$\ds \limsup_{n \mathop \to \infty} \expect {X_n \mid \GG} \le \expect {\limsup_{n \mathop \to \infty} X_n \mid \GG}$ almost surely.

$\blacksquare$

Source of Name

This entry was named for Pierre Joseph Louis Fatou.