Applying Proposition 1 to the unconditional Strauss process,
any 
with 
is a direction of recession. But
if 
then 
and the limit obtained using the
proposition is the uninteresting process that has no points with probability
one.
If 
and 
, then
The process obtained by conditioning a Strauss process on 
is
the general hard core process. Note that one obtains the same hard
core process
by conditioning any Strauss process 
for any
.
Thus we obtain the complete family of hard core processes by conditioning
Strauss processes 
with 
, that is, Poisson processes.
This connexion between the Strauss process and hard core process is,
of course, well-known, though not usually derived using
general properties of exponential families. The reason for
drawing the connexion here, is that we will also be interested in
other point process models for which there is no existing literature.
The point of the proposition is that whenever we consider a point process
model that is an exponential family a natural question arises as to
what models arise from taking limits in directions of recession using
Proposition 1.
There is still one kind of direction of recession left to consider.
If 
and , then the characterization of
is not obvious. Every such vector is a direction of recession.
To see this we divide S into m disjoint subregions of radius less than
r so that any two points in a subregion are neighbours. If there are
points in the ith subregion, then 
and
. It is easy to see that the latter is
minimized over a real ntuples 
when 
.
Hence
Since this is quadratic in n(x), it follows that
is bounded on 
.
Thus , the set of x where the least upper bound is achieved,
is nonempty, and a limit process in the direction can be defined.
What it not clear, is what the process or the set looks like.
Perhaps this is related to the strange plots of unconditional Strauss processes
exhibited by Professor Møller at the meeting.