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Conditional Expectation example

Consider the following probability space \((\Omega,P,\mathcal{F})\) where \(\Omega=\big\{ (\omega_0,\omega_1) : \omega_i \in \{-1,0,1\} \big\}\). Take of each of the \(\omega_i\) to be mutually independent with \(\mathbf P(\omega_i=0)=\frac12\) and \(\mathbf P(\omega_i=\pm 1)=\frac14\). (\(\mathcal{F}\) is just the \(\sigma\)-algebra generated by the collection of single points, but this is not important)

For \(n=0\) or \(1\) define the random variables \(X_n\) by  \(X_0=\omega_0\) and \(X_1=\omega_0 \omega_1\).

  1. What is \(\mathbf P(X_1=1)\) ? What is \(\mathbf E X_0\) ?
  2. Let \(A\) be the event that \(\{X_1\neq 0 \}\). What is \(\sigma(A)\) ?
  3. What is \(\sigma(X_0)\)( and \(\sigma(X_1)\) ?
  4. What is \(\mathbf E(X_1 | A)\) ? What is \(E(X_1 | X_0)\) ?

 

First Borel-Cantelli Lemma

The first Borel-Cantelli lemma is the principle means by which information about expectations can be converted into almost sure information.

 

Let \((\Omega, {\cal F}, P)\) be a probability space and let \(A_k\), \(k=1,2,3,…\) by a sequence of events in \({\cal F}\) for which
\[
\sum_{k=1}^{\infty} P(A_k) < \infty
\]
Prove the (first) Borell-Cantelli lemma which says that
\[
P\Big\{\bigcap_{N=1}^{\infty}\bigcup_{k=N}^{\infty} A_k \Big\}=0
\]
Interpret this event whose probability is zero. (Say sometime like “it is the event in which all of the events \(A_k\) happen”. This of course not the right answer.)
Now let us use this result: Let \(X_k\) be a sequence of random variables and suppose that there is another random variable \(X\) such that if
\[
A_k =\{\omega\in\Omega ~|~ ~~ |X_k(\omega)-X(\omega)|>\delta \}
\]
then
\[
\sum_{k=1}^{\infty} P(A_k) < \infty
\]
for all \(\delta >0\). Show that \(X_n \to X\) with probability one. (That is the set of \(\omega\) so that \(X_n(\omega) \to X(\omega)\) has measure (probability) one.)

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