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Simple Numerical Exercise
Let \(\omega_i\) , \(i=1,\cdots\) be a collection of mutually independent, uniform on \([0,1]\) random variables. Define
\[\eta_i(\omega)= \omega_i -\frac12\]
and
\[X_n(\omega) = \sum_{i=1}^n \eta_i(\omega)\,.\]
- What is \(\mathbf{E}\,X_n\) ?
- What is \(\mathrm{Var}(X_n)\) ?
- What is \(\mathbf{E}\,X_{n+k} | X_n \) for \(n, k >0\) ?
- What is \(\mathbf{E}(\,X_5^2 \,|\, X_3)\) ?
- [optional] Write a computer program to simulate some realizations of this process viewing \(n\) as time. Plot some plots of \(n\) vs \(X_n\).
- [optional] How do you simulations agree with the first two parts ?
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.)
Chebychev Inequality
Let \(X\) be a real valued random variable. Prove the Chebychev inequality:
\[
\mathbf{P}\{|X| \geq \lambda\} \leq \frac{\mathbf{E}\{|X|^p \}}{\lambda^p }
\]
for all \(1\leq p <\infty\) as well as its exponential form
\[
\mathbf{P}\{|X| \geq \lambda\} \leq \frac{\mathbf{E}\{e^{k|X|} \}}{e^{k\lambda} }
\]
when the integrals on the right are finite. Here \(k\) and \(\lambda\) are positive constants.
Hint: combine the facts that \[\mathbf{P}\{|X|^p \geq \lambda\} = \mathbf{E} \mathbf{1}_{\{|X|^p \geq \lambda\}}\]
and
\[ \mathbf{1}_{\{|X|^p \geq \lambda\}}\lambda^p \leq \mathbf{1}_{\{|X|^p \geq \lambda\}}|X|^p\] and \[1= \mathbf{1}_{\{|X|^p \geq \lambda\} }+ \mathbf{1}_{\{|X|^p <\lambda\}}\, . \]
