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{\bf HW 5, due Monday June 7}
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Chapter 26 Problems 30, 32, 33 and
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(4) Let $X_{n}$ be a branching process with expected number
of offsprings of each member of population $>1$, so that
$c:=P^{1}(\tau_{0}<\infty)<1$.
Prove that $\lim_{n\to\infty} c^{X_{n}}$ exists with probability 
1 and conclude that
 on the set $\{\tau=\infty\}$ we have $\lim_{n\to\infty}
X_{n}=\infty$ (a.s.). (Hint: use that $c^{x}$ is harmonic
and use Theorem 24-19)


(5) Let $Z_{n}=(X_{n},Y_{n})$ be
defined by $X_{0}=Y_{0}=0$, $X_{n}=\xi_{1}+...+\xi_{n}$ and
$Y_{n}=\eta_{1}+...+\eta_{n}$ for $n\geq1$, where
$\xi_{n},\eta_{n}$ are iid with $P\{\xi_{n}=\pm1\}=
P\{\eta_{n}=\pm1\}=1/2$. Prove that $Z_{n}$ is
a Markov sequence on the state
space $\mathbb{Z}^{2}$ (two dimensional lattice) with transition
probabilities given by
$$
P^{(x,y)}(x+1,y+1)=P^{(x,y)}(x-1,y-1)=\frac{1}{4},
$$
and
$$
P^{(x,y)}(x+1,y-1)=
P^{(x,y)}(x-1,y+1)=\frac{1}{4}.
$$

(6) (Recurrence of two-dimensional random walk)
For the process from (5), by remembering Bernoulli, prove that
$$
u_{2n}:=P^{(0,0)}\{Z_{2n}=0\}=4^{-2n}\binom{2n}{n}^{2}.
$$
By using Stirling's formula show that $\sum_{n}u_{2n}=\infty$.
On the basis of renewal theory derive from here that the
number of returns of $Z_{n}$ to the origin is infinite with probability
one.

(7) Let $w_{t}$ be a Wiener process. Prove that $\int_{0}^{1}
w_{t}\,dt$ is a Gaussian random variable and find its parameters.
({\em Hint\/}: You may need to use $(\int_{0}^{1}
w_{t}\,dt)^{2}=\int_{0}^{1}\int_{0}^{1}w_{t}w_{s}\,dsdt$.
It is also a good idea to represent $\int_{0}^{1}
w_{t}\,dt$ as a limit of partial sums.)

(8) (Self-similarity of the Wiener process)
Let $w_{t}$ be a Wiener process and $c\ne0$ be a constant.
Prove that $cw_{t/c^{2}}$ is again a Wiener process.

(9) 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 $w_{t}$ is a martingale, prove that
$$
P\{w_{\tau(a,b)}=b\}=\frac{|a|}{|a|+b}.
$$



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