Solutions
to
Homework
9
6.262
Discrete
Stochastic
Processes
MIT,
Spring
2011
Exercise
5.6
:
Let
{
X
n
;
n
≥
0
}
be
a
branching
process
with
=
1.
X
0
Let
Y
,
σ
2
be
the
mean
and
variance
of
the
number
of
oﬀspring
of
an
individual.
a)
Argue
that
lim
n
X
n
exists
with
probability
1
and
either
has
the
value
0
(with
→∞
probability
F
10
(
∞
))
or
the
value
∞
(with
probability
1
−
F
10
(
∞
)).
Solution
5.6a
We
consider
2
special,
rather
trivial,
cases
before
considering
the
impor-
tant
case
(the
case
covered
in
the
text).
Let
p
i
be
the
PMF
of
the
number
of
oﬀspring
of
each
individual.
Then
if
p
1
=
1,
we
see
that
X
n
=
1
for
all
n
,
so
the
statement
to
be
argued
is
simply
false.
It
is
curious
that
this
exercise
has
been
given
many
times
over
the
years
with
no
one
pointing
this
out.
The
next
special
case
is
where
p
0
=
0
and
p
1
<
1.
Then
X
n
+1
≥
X
n
(
i.e.
,
the
population
never
shrinks
but
can
grow).
Since
X
n
(
ω
)
is
nondecreasing
for
each
sample
path,
either
lim
n
X
n
(
ω
) =
∞
or
lim
n
X
n
(
ω
) =
j
for
some
j <
∞
.
The
latter
case
is
impossible,
→∞
→∞
since
mj
P
jj
=
j
p
1
and
thus
P
m
jj
=
p
1
→
0.
Ruling
out
these
two
trivial
cases,
we
have
p
0
>
0
and
p
1
<
1
−
p
0
.
In
this
case,
state
0
is
recurrent
(
i.e.
,
it
is
a
trapping
state)
and
states
1
,
2
,...,
are
in
a
transient
class.
To
see
this,
note
that
P
10
=
p
0
>
0,
so
F
11
(
∞
)
≤
1
−
p
0
<
1,
which
means
by
de±nition
that
state
1
is
transient.
All
states
i >
1
communicate
with
state
1,
so
by
Theorem
5.1.1,
all
states
j
≥
1
are
transient.
Thus
one
can
argue
that
the
process
has
‘no
place
to
go’
other
than
0
or
∞
.
The
following
ugly
analysis
makes
this
precise.
Note
from
Lemma
5.1.1
part
4
that
lim
P
t
jj
=
.
t
→∞
n
∞
≤
t
Since
this
sum
is
nondecreasing
in
t
,
the
limit
must
exist
and
the
limit
must
be
±nite
This
means
that
lim
P
n
jj
= 0
±
t
n t
→∞
≥
Now
we
can
write
P
n
1
j
=
n
f
P
j
n
−
n
≤
1
j
j
,
from
which
it
can
be
seen
that
lim
t
P
=
→∞
n
≥
t
1
j
0.
From
this,
we
see
that
for
every
±nite
integer
,
±
lim
P
n
1
j
= 0
t
→∞
n
≥
t j
=1
This
says
that
for
every
± >
0,
there
is
a
t
suﬃciently
large
that
the
probability
of
ever
entering
states
1
to
on
or
after
step
t
is
less
than
±
.
Since
± >
0
is
arbitrary,
all
sample
paths
(other
than
a
set
of
probability
0)
never
enter
states
1
to
after
some
±nite
time.
1

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