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\begin{document}

\centerline {\bf Math 563 - Fall 15 - Homework 3} 

\smallskip

\no 1. (from Resnick, slightly modified) Let $X_n$ be a sequence of 
RV's on $(\Omega,\field,P)$. Define 
\bea
S_n=\sum_{i=1}^n X_i
\eea
Let $\tau= \inf \{n>0: S_n>0\}$. Note that $\tau$ may sometimes be 
$\infty$. 

\no (a) Prove $\tau$ is a random variable, i.e., it is measurable.
Note that $\tau$ takes values in the extended reals, so by measurable
I mean measurable with respect to the Borel sets in the 
extended reals. 

\no (b) Define $X$ to be $S_\tau$ when $\tau$ is finite and $0$
when $\tau=\infty$. Prove $X$ is a random variable. 

\smallskip

\no 2. Let $A_n$ be independent events. Prove that 
\bea
P(\cap_{n=1}^\infty A_n) = \prod_{n=1}^\infty P(A_n)
\eea

\smallskip


\no 3. (from Durrett) Let $\Omega=(0,1)$, $\field$ be the Borel sets
in $(0,1)$, and let $P$ be Lebesgue measure. 
Define RV's by 
\bea
X_n(\omega)=\cases{
0 & if $[2^n \omega]$ is even \cr
1 & if $[2^n \omega]$ is odd \cr}
\eea
where $[x]$ is the largest integer less than or equal to $x$. 
Prove that $\{X_n\}_{n=1}^\infty$ are independent random variables.
Note that this gives a rigorous construction of the probability 
space for flipping a fair coin infinitely many times.

\smallskip

\no 4. Prove that a random variable $X$ is independent of itself if 
and only if there is a constant $c$ such that $P(X=c)=1$. 
Hint: What can you say about the distribution function of $X$?

\no {Optional challenge:} Prove that a random element
$X: \Omega \ra \Omega^\prime$ is independent of itself if and only if 
there is a $c \in \Omega^\prime$ such that $P(X=c)=1$. 
Warning: I don't know how to do this. 

\smallskip

\no 5. (from Resnick) Let $A_n$ be a sequence of independent events
with \\
$P(A_n)<1$ for all $n$. Prove 
\bea
P(\cup_{n=1}^\infty A_n)=1 \, if \, and \, only \, if 
P(A_n \, i.o.)=1
\eea
Give a counterexample to show that the condition $P(A_n)<1$ cannot
be dropped.

\smallskip

\no 6. (from Resnick) 
Let $X_n$ be a sequence of independent random variables. 
Prove
\bea
P(\sup_n X_n < \infty) =1
\eea
if and only if there is a finite constant $M$ such that 
\bea
\sum_{n=1}^\infty P(X_n >M) < \infty
\eea

\end{document}
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