Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

334 Actuarial Modelling of Claim Counts
hold true, with the convention that the functions take the value 0 where they have not been
defined.
Proof
It is trivial that for t = 1 we have
−
x
fx 0 = e x>0
x!
f0 1 = e −
As f ⋆t is the t-fold convolution of a lattice random vector, we are in a position to apply the
bivariate extension of De Pril’s algorithm given in Property 9.2. As that algorithm needs a
mass at the origin, we define an auxilliary probability mass function
We find
g ⋆t 0 0 = e −t
g ⋆t x y = e ( x∑
u=0
+
x∑
u=1
g ⋆t x y = f ⋆t x t − y
( t + 1
x u − 1 )
g ⋆t x − u y − 1gu 1
( t + 1
x u − 1 )
g ⋆t x − u ygu 0
)
x≥ 1
( x∑ ( ) t + 1
g ⋆t x y = e − 1 g ⋆t x − u y − 1gu 1
u=0
y
)
x∑
+ −1g ⋆t x − u ygu 0 y≥ 1
u=1
Because g0 1 = 0 and gx 0 = 0 for x>0, we obtain
g ⋆t 0 0 = e −t
( )
x∑ t + 1
g ⋆t x y = e
u=1
x u − 1 g ⋆t x − u y − 1gu 1 x ≥ 1
( )
x∑ t + 1
g ⋆t x y = e − 1 g ⋆t x − u y − 1gu 1 y ≥ 1
y
u=1
Because g ⋆t x 0 = 0 for x>0, only the second recursive formula has to be used. This
formula is numerically stable because t + 1/y − 1 > 0.
□
In order to obtain the unconditional probability mass function
f ⋆t x y = PrN • = x I • = y