HW 5, due Monday June 8
Chapter 26 Problems 30, 32, 33 and
(4) Let 
be a branching process with expected number
of offsprings of each member of population >1, so that
.
Prove that 
exists with probability
1 and conclude that
on the set 
we have 
(a.s.). (Hint: use that 
is harmonic
and use Theorem 24-19)
(5) Let 
be
defined by 
, 
and
for 
, where
are iid with 
. Prove that 
is
a Markov sequence on the state
space 
(two dimensional lattice) with transition
probabilities given by
and
(6) (Recurrence of two-dimensional random walk) For the process from (5), by remembering Bernoulli, prove that
By using Stirling's formula show that 
.
On the basis of renewal theory derive from here that the
number of returns of to the origin is infinite with probability
one.
(7) Let 
be a Wiener process. Prove that 
is a Gaussian random variable and find its parameters.
(Hint: You may need to use 
)
(8) (Self-similarity of the Wiener process)
Let be a Wiener process and 
be a constant.
Prove that 
is again a Wiener process.
(9) Let be a Wiener process, 
. We know that
. In particular,
where 
and
is the first time hits either 
or 
.
By using that is a martingale, prove that
