\documentclass{amsart}
\renewcommand{\thepage}{}
\begin{document}


\centerline{\bf Take home final}\bigskip

1. Give an example of two sequences of nonnegative
numbers $u_{n},n\geq1$ and $v_{n},n\geq1$ such that
$$
\sum_{n=1}^{\infty}s^{n}u_{n}=
\sum_{n=1}^{\infty}s^{n}v_{n}\quad\forall s\geq0
$$
and, for any $n\geq1$, we have $u_{n}\ne v_{n}$.

2. Explain why the function defined by $(1-v^{2})^{1/2}$ for $|v|\leq1$
and $0$ for $|v|\geq1$ is not a characteristic function.

3. Give an example of iid $X_{1},X_{2},...$ such that
$(X_{1}+...+X_{n})/n$ converges in distribution but not
in probability. (Hint: upon assuming that
$(X_{1}+...+X_{n})/n$ tends in probability to a random variable
$\xi$, prove that $\xi$ is constant (a.s.) due to
Kolmogorov's 0-1 law.)

4. Give an example of a sequence of moment generating functions
$\varphi_{n}(u)$ such that $\lim_{n\to\infty}\varphi_{n}(u)$
exists  for any $u\geq0$, but this limit is not a continuous
function of $u\in[0,\infty)$. 

5. Let $(X,Y,Z)$ be a 3 dimensional normal variable. Prove that
$X,Y,Z$ are independent iff they are pairwise uncorrelated.

6. Let $w_{t}$ be a Wiener process, $\tau_{a}=
\inf\{t\geq0:w_{t}=a\}$. We know that
$P\{\tau_{a}<\infty\}=1$. In particular,
$P\{\tau(a,b)<\infty\}=1$ where $\,a<0<b\,$ and
$\tau(a,b)$ is the first time $w_{t}$ hits either $\,a\,$ or $\,b\,$.
By using that $w^{2}_{t}-t$ is a martingale, prove that
$$
E\tau(a,b)=|a|b.
$$ 

7. For any constant $b_{}$ and $a_{},T_{}>0$ define $$P_{T}(a,b)=
P\{\max_{s\leq T}(w_{s}+bs)<a\}.$$  We know that
$P_{T}(a,b)=$
$$
\frac{1}{\sqrt{2\pi T}}\int_{-\infty}^{-bT}
e^{-\frac{1}{2T}(x+a)^{2}}\,dx
-e^{2ba}
\frac{1}{\sqrt{2\pi T}}\int_{-\infty}^{-bT}
e^{-\frac{1}{2T}(x-a)^{2}}\,dx.
$$
By using change of variables show that
$P_{T}(a,b)=$
$$\frac{1}{\sqrt{2\pi }}\int_{-\infty}^{a/\sqrt{T}-b\sqrt{T}}
e^{-\frac{1}{2}x^{2}}\,dx
-e^{2ba}
\frac{1}{\sqrt{2\pi }}\int_{-\infty}^{-a/\sqrt{T}-b\sqrt{T}}
e^{-\frac{1}{2}x^{2}}\,dx.
$$
By letting $T\to\infty$ conclude that
$P\{\max_{s<\infty}(w_{s}+bs)<a\}$ equals 0 if $b_{}\geq0$
and $1-e^{2ba}$ if $b_{}<0$. (Warning: the case $b_{}=0$ requires
a little bit extra attention.)
\end{document}
