We are now in a position to use the theorem of the previous section to establish the following theorem.
If 
is concentrated on a single point then only one trajectory is
possible and
assigns probability 1 to this trajectory maximizing the
likelihood.
We assume that has lattice size 1; we have not pursued the
details for
other lattice sizes, though we believe the additional complications
are not serious.
We are also confident that the restriction that 
not be an
integer is
inessential but have not been able to prove it.
The theorem is proved in several steps. First we use the fact that
is the exact mle of 
to prove that we can maximize over
distributions
whose mean is not very different from that of 
. When
the mean of
is not an integer all such 
have variances bounded away from 0
and we can restrict
our attention to distributions in 
for some
positive 
.
For such the likelihood can be approximated by a product of
two
terms. The first of these two terms arises from the normal
approximation to
the distribution of 
. This term can be maximized explicitly over
the class of
distributions whose variance differs by more than
from that of .
The maximum value will then be shown to be
smaller than the corresponding term in the approximation for the
likelihood under
. The second of the two
terms arises from the distribution of the residue classes 
.
Under and any
other single whose lattice size is 1 the sequence is
asymptotically iid
and uniform on the set of possible residue classes. For distributions
whose
lattice size is not 1 we can show that almost every (for )
trajectory eventually
has an initial segment whose probability is 0. The problem
remaining is one of
uniformity. For any 
there will be which have lattice size 1
but are very
close to a distribution whose lattice size is larger than 1. For such
the
likelihood of the initial segment will not be exactly 0 but at the
same time the
sequence will not be approximately uniform under .
Most of the notational difficulty in what follows is dedicated to
dealing with such
.
We will use the notation 
so that the
likelihood is 
. Let 
be 
standardized for . In what
follows
with 
integers and 
.
Throughout, lower case letters will be used to denote observed
values of such
random
variables as 
, 
and so on. Our approximation to 
will
be
where
and
.
We will need the following facts about branching processes. References may be found in Guttorp(1991).
almost surely.
for which
(Keiding and Lauritzen, 1978).
almost surely.
almost surely.
