Another standard property of exponential families gives new processes by
taking limits. Let 
be a direction in the parameter space such
that 
for all 
and for
all s ;SPMgt; 0. Then is called a direction of recession
of 
(Rockafellar, 1970, p. 61). If is closed, then
is a direction of recession if there exists even one
such that for all s ;SPMgt; 0
(Rockafellar, 1970, Theorem 8.3). For any direction , let
be the essential supremum of the natural statistic t(X),
that is the infimum of all real numbers r such that
(since the distributions in the family are
absolutely continuous with respect to each other, any
can be used here). If is finite, let
Define normalized densities and measures by (1.14) and
(1.15)
with 
and if 
define
Then 
is a density with respect to 
of the conditional probability measure 
. (By
definition of essential supremum and continuity of measure, the
set where is 
has -measure zero.)
This proposition follows from Barndorff-Nielsen (1978), p. 105.
