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\begin{document}

\centerline {\bf Math 563 - Fall '14 - Homework 6} 

\smallskip

\medskip

\no 1. (from Durrett) Let $X_n$ be a sequence of integer valued random 
variables, $X$ another integer valued random variable. Prove that 
$X_n$ converge to $X$ in distribution if and only if 
\bea
\lim_{n \ra \infty} P(X_n=m) = P(X=m)
\nonumber
\eea
for all integers $m$. 

\medskip

\no 2. 
Take a coin with the probability of 
heads equal to $p \in (0,1)$ and flip it $n$ times. Let $X$
be the number of heads you get. The distribution of $X$ is 
called the binomial distribution with $n$ trials. It is easy to 
show that 
$$
P(X=k)= {n \choose k} p^k (1-p)^{n-k} , \quad k=0,1,2, \cdots n
$$

Now fix a $\lambda>0$, and let $X_n$ have a binomial distribution 
with $n$ trials and $p=\lambda/n$. 
Prove that as $n \rightarrow \infty$, $X_n \Rightarrow X$ where $X$ has the 
Poisson distribution with parameter $\lambda$, i.e., 
$$
P(X=k)= e^{-\lambda} \frac{\lambda^k}{k!}, \quad k=0,1,2,\cdots
$$


\medskip

\no 3. (a) Let $\mu_n$ be a sequence of probability measures which have 
densities $f_n(x)$ with respect to Lebesgue measure. Suppose that 
$f_n(x) \ra f(x)$ a.e. where $f(x)$ is a density, i.e., a non-negative
function with integral $1$. Prove that $\mu_n$ converges in 
distribution to $\mu$ where $\mu$ is $f(x)$ times Lebesgue measure. 

\no (b) 
Suppose $F_n$ has density $1-\cos(2n\pi x)$ on $[0,1]$ and the density is 
zero outside this interval. 
Show that  $F_n$ converges weakly to the uniform distribution on $[0, 1]$,
but that the densities $f_n$ do not converge.

\medskip

\no 4. Suppose that the random variables $X_n$ are defined on the 
same probability space and there is a constant $c$ such that $X_n$ 
converges in distribution to the random variable $c$. Prove 
or disprove each of the following

\no (a) $X_n$ converges to $c$ in probability

\no (b) $X_n$ converges to $c$ a.s.


\medskip

\no 5. (from Durrett, converging together lemma)
Suppose $X_n \Rightarrow X$ and $Y_n \Rightarrow c$ where $c$ is a constant. 
Prove that $X_n + Y_n \Rightarrow X +c$. 
Note that this implies that if $X_n \Rightarrow X$ 
and $Y_n - X_n \Rightarrow 0$, then $Y_n \Rightarrow X$. 

\medskip

\no 6. 
Let $X_n, Y_n, X, Y$ be random variables defined on the same
probability space.  
Suppose that $X_n$ converges in distribution to $X$ and $Y_n$ 
converges in distribution to $Y$. Suppose further that for each $n$,
$X_n$ and $Y_n$ are independent, and that $X$ and $Y$ are independent. 
Prove that $X_n + Y_n$ converges in distribution to $X+Y$. 

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