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\centerline {\bf Math 563 - Fall '14 - Homework 3} 

\smallskip


\no 1. Let $\Omega=\{a,b,c,d\}$. Let $\f$ be the $\sigma$-field 
containing all subsets of $\Omega$. Let $\e=\{\{a,b\},\{b,c\}\}$. 
Show that $\sigma(\e)=\f$ but there exist two different probability 
measures on $(\Omega,\f)$ which agree on $\e$. 

\medskip

\no 2. (from Durrett) 

\no (a) Let $X$ and $Y$ be independent random variables which 
take values in the integers. Prove that the distribution of $X+Y$ 
is given by 
$$
P(X+Y=n) = \sum_{m= -\infty}^\infty P(X=m) P(Y=n-m)
$$

\no (b) $X$ has a Poisson distribution with parameter $\lambda>0$
if it takes on the values $0,1,2,\cdots$ and 
$$
P(X=n)= \frac{e^{-\lambda} \lambda^n}{n!}
$$
Show that if $X$ and $Y$ are independent random variables, $X$ has 
a Poisson distribution with parameter $\lambda$ and 
$Y$ has a Poisson distribution with parameter $\mu$, then $X+Y$
has a Poisson distribution. What is the parameter for $X+Y$ ?

\medskip

\no 3. Let $X_1, \cdots, X_n$ be random variables and let 
$Y_1, \cdots, Y_n$ be random variables. Suppose that their
joint distribution functions are equal:
$$
F_{X_1,\cdots,X_n}(x_1,\cdots,x_n)=F_{Y_1,\cdots,Y_n}(x_1,\cdots,x_n)
$$
Use the $\pi-\lambda$ theorem to prove that their 
joint distributions are equal, i.e., 
$$
\mu_{X_1,\cdots,X_n}=\mu_{Y_1,\cdots,Y_n}
$$
where the two measures are Borel measures on $\reals^n$.
There is a theorem that says a Borel measure on $\reals^n$ is determined 
by its values on the rectangles. You should not use this theorem.

\medskip

\no 4. Suppose $X_n$ are identically distributed, uncorrelated, and have 
finite second moment, i.e.,
$EX_n^2 < \infty$. We proved a weak law of large numbers 
which says that for any
$\epsilon > 0$, the probability 
$$P(|{1 \over n} \sum_{k=1}^n X_k - \mu| > \epsilon)$$
converges to zero as $n \rightarrow \infty$, but we didn't say anything 
about how fast it converges.

\no (a) Use the proof from class of the weak law for the
case of finite second moment to show it converges to zero at least 
as fast as $1/n^p$ for some power $p$. (I.e., show that the probability 
is $\le c/n^p$ for some constant $c$, which can depend on $\epsilon$.)
You should find the biggest $p$ you can. 

\no (b) Now suppose that you also know that $EX_n^4 < \infty$ and 
the $X_n$ are independent. 
Prove that $P(|{1 \over n} \sum_{k=1}^n X_k - \mu| > \epsilon)$
converges to zero faster than your bound from (a), i.e., with a bigger $p$. 

\medskip

\no 5. (from Kallenburg) 

\no (a) 
Let $X_n$ be an independent sequence of random variables which take
values in $[0,1]$. 
Prove that 
$$
E[\prod_{n=1}^\infty X_n] = \prod_{n=1}^\infty E X_n
$$

\no (b) 
Let $A_n$ be an independent sequence of events
Prove that 
$$
P(\cap_{n=1}^\infty A_n) = \prod_{n=1}^\infty P(A_n)
$$

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