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\begin{document}

\centerline {\bf Math 563 - Fall '15 - Homework 7} 

\smallskip
  
\no 1. The Cauchy distribution has the density function
$$
f(x) =\frac{1}{\pi (1+x^2)}
$$
I.e., the distribution is Lebesgue measure on the real line times
this function.

\no (a) Compute the characteristic function of a Cauchy random variable. 
Hint: contour integration.

\no (b) Show that if $X_n$ is an i.i.d. sequence with the Cauchy distribution,
then for all $n$, $\frac{1}{n}\sum_{k=1}^n X_k$ has the Cauchy distribution. 

\medskip

\no 2. Suppose that for each $n$, $X_n$ has a normal distribution with 
mean $0$ and variance $\sigma^2_n$. Suppose there is a random variable 
$X$ such that $X_n \Rightarrow X$. Prove $\sigma_n^2$ converges to 
some constant $\sigma \in [0,\infty)$ and that $X$ is normal with 
variance $\sigma^2$. (Normal with variance $0$ should be interpretted 
to mean that $X=0$.)

\medskip

\no 3. {\bf Renewal theory} (from Resnick) 
Suppose $X_n$ are i.d.d. non-negative random variables. 
Let $\mu$ be their mean and $\sigma^2$ their variance. Let 
\bea
N(t)=\sup \{n: S_n \le t\}
\eea
Prove that 
\bea
\frac{N(t)-\frac{t}{\mu}}{\sigma t^{1/2} \mu^{-3/2}} \Rightarrow Z
\eea
where $Z$ has a standard normal distribution.

\medskip

\no 4. (from Durrett) Let $X_n$ be independent, and suppose there is 
a constant $M$ such that $|X_n| \le M \, a.s.$ for all $n$. Suppose also 
that $\sum_n var(X_n)=\infty$. Let $S_n=X_1+ \cdots + X_n$. 
Prove that 
\bea
\frac{S_n -E S_n}{\sqrt{var(S_n)}} \Rightarrow Z
\eea
where $Z$ has a standard normal distribution. 

\medskip

\no 5. The gamma distribution is a two parameter family of densities.
The parameters $\lambda$ and $w$ are both positive.  
The density is 
$$
f(x)=\frac{\lambda^w}{\Gamma(w)} x^{w-1} \exp(-\lambda x) 
$$ 
for $x \ge 0$. The density is $0$ for $x<0$.
The gamma function is defined by 
$$
\Gamma(w) = \int_0^\infty \, x^{w-1} \, e^{-x} \, dx 
$$
Integration by parts shows $\Gamma(w+1)=w \Gamma(w)$. If $w$ is 
an integer then $\Gamma(w)=(w-1)!$. A little calculus shows the mean 
of the gamma distribution is $w/\lambda$ and the variance is $w/\lambda^2$. 

\smallskip
\no (a) Find the characteristic function of the gamma distribution. 


\smallskip
\no (b) For positive integers $n$ let $X_n$ be a random variable with a 
gamma distribution with $w=n$ and $\lambda=1$. Prove that
$(X_n - n)/\sqrt{n}$ converges in distribution to a standard normal. 

\end{document}
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