We have now reduced the proof of theorem 3 to that of establishing
almost surely. We will prove this by establishing
almost surely and
almost surely.
To prove 9 note that 
differs from
by a constant not depending on 
(note that we suppress the
dependence
of 
and 
on ). For fixed this is maximized
as a
function of by 
. The profile obtained by replacing
by
is concave and is maximized over all by 
.
Since
this estimate is strongly consistent on the explosion set, the profile
is
maximized, subject to 
at one of
the two
points 
. Finally, 9 is
easily established by considering the two values individually.
Fix 
and let 
be the event that 
for
all 
. Then 
and 
is the explosion
set except for a 
-null event. For trajectories in 
with
chosen large we write
We will choose 
and 
depending on . In particular
let
with 
to be chosen later and
Each 
is a probability so bounded by 1. Thus
The first term in 12 is free of and after
division by 
converges to 0 almost
surely on . Also on 
grows exponentially
when 
and so
. From 1
we obtain
On 
we have 
. Thus, except for a fixed number of 
(depending only on and 
) the right hand side of 13 is less than 
permitting a Taylor expansion of the 
. Hence
almost surely.
For the fourth term in 12 we have
for all 
. Letting 
a Taylor expansion of the logarithm then shows that
which converges to 0 almost surely.
It remains to deal with the second term in 12.
This
is
the most technical part of the argument.
Consider now any for which 
. (Remember that
depends on .) There is an 
for which
. There is then a constant 
not
depending
on or 
such that for all small
Since 
is in the convex hull of the points 
as 
runs over 
and since this convex
hull is a regular polygon there is a constant 
depending only on
such that there must exist an 
with
Finally there is a 
depending only on such that
Let 
be the set of all such that 15
holds. Note that either 
and the second term above is 0 or
belongs to some . For in
and 
we have
Let
Then
An argument like that at (14)
shows that the sequence 
is asymptotically
an iid sequence of uniformly distributed random variables. Using this argument we can establish
Since 
we can choose so that
With this choice of we can use the lemma to prove
Theorem 3 follows.
