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.