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218

P. R. Bouzas, N. Ruiz-Fuentes

!

t

t

0

λ

(

σ, x

(

σ

))

. 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

))

%

n

exp

#

$

t

t

0

λ

(

σ, x

(

σ

))

% &

=

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 moment-generating 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

))

-.

=

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

.