218
P. R. Bouzas, N. RuizFuentes
!
t
t
0
λ
(
σ, x
(
σ
))
dσ
. As previously mentioned, then, it is clear that the mean is
also a stochastic process and that the CP can be defined using the mean pro
cess.
The statistics of the CP are obtained by the usual conditioning method ex
pressed in terms of the intensity process. The expression of the probability mass
function in terms of the mean process was provided in Bouzas et al. (2002) and
the characteristic function derived in Bouzas et al. (2006a).
•
Probability mass function
Following the notation in Snyder and Miller (1991),
P
[
N
(
t
) =
n
] =
E
{
P r
[
N
(
t
) =
n
]

x
(
σ
) :
t
0
≤
σ > t
}
.
This notation will be used from now on.
Using the conditioning method, the expression of the probability mass
function takes the form
P
[
N
(
t
) =
n
] =
E
"
1
n
!
#$
t
t
0
λ
(
σ, x
(
σ
))
dσ
%
n
exp
#
−
$
t
t
0
λ
(
σ, x
(
σ
))
dσ
% &
=
1
n
!
E
'
Λ(
t, x
(
t
))
n
e
−
Λ(
t,x
(
t
))
(
=
1
n
!
G
n
)
Λ(
t,x
(
t
))
(
−
1)
for
n
= 0
,
1
,
2
, . . .
and where
G
Λ(
t,x
(
t
))
(
s
) is the momentgenerating func
tion of Λ(
t, x
(
t
)).
•
Characteristic function
M
N
(
t
)
(
iu
) =
E
)
exp
* +
e
iu
−
1
, $
t
t
0
λ
(
σ, x
(
σ
))
dσ
.
=
E
/
exp
0+
e
iu
−
1
,
Λ(
t, x
(
t
))
12
=
M
Λ(
t,x
(
t
))
(
e
iu
−
1)
where
M
Λ(
t,x
(
t
))
(
s
) is the characteristic function of Λ(
t, x
(
t
)).
As a consequence, it is obtained that
E
[
N
(
t
)] =
E
[Λ(
t, x
(
t
))] and
V ar
[
N
(
t
)] =
V ar
[Λ(
t, x
(
t
))] +
E
[Λ(
t, x
(
t
))]
2.2. Compound Cox process
The mathematical formal definition of a compound Cox process (CCP) is the
following. A CCP is a CP
{
N
(
t
);
t
≥
t
0
}
with intensity
{
λ
(
t, x
(
t
));
t
≥
t
0
}
where
{
x
(
t
);
t
≥
t
0
}
is the information process and with i.i.d. marks associated to the
arrival times and independent of the point process, where the
n
th arrival time
will be denoted by
w
n
and its mark by
u
n
which is a realization of the random
variable
U
in
U
.