124 B. Sundt Insurance: Mathematics and Economics 27 2000 123–136
extension of Sundt 1999a to the recursion of Sundt 1992. We show that, like in the univariate case, the multivariate De Pril transform is additive for convolutions, and discuss De Pril transforms of compound functions, higher order
functions, and infinitely divisible distributions. We also express the De Pril transform of the distribution of a vector of independent sub-vectors of random variables by the De Pril transforms of the marginal distributions of these
sub-vectors. Finally, in Section 4 we introduce the Dhaene–De Pril transform as a multivariate version of a function discussed by Dhaene and De Pril 1994 in the univariate case, and discuss its advantages and disadvantages
compared to the De Pril transform.
Sundt 1999b discusses some results related to the present paper. 1D. In the present paper we shall work on distributions in the representation of their probability functions.
Therefore, we shall for simplicity refer to probability functions as distributions.
2. The univariate De Pril transform
2A. Let N
1
denote the set of non-negative integers and P
10
the class of distributions on N
1
with a positive mass at zero. For a positive integer k and functions a and b we define R
k
[a, b] to be the distribution p ∈ P
10
given by the recursion
pn =
k
X
i=1
ai + bi
n pn − i
n = 1, 2, . . . , 2.1
with pn = 0 for all n 0. Following Sundt 1992, we let R
k
denote the class of all such distributions for fixed k; we allow k = ∞. We have that R
∞
= P
10
. For simplicity, we let ai = bi = 0 for i = k + 1, k + 2, . . . 2B. Let N
1+
denote the set of positive integers and P
1+
the class of distributions on N
1+
. For p ∈ P
10
and h ∈ P
1+
, the compound distribution p ∨ h is given by p ∨ h =
∞
X
n=0
pnh
n∗
. 2.2
As h
n∗
x = 0 x ∈ N
1
; n = x + 1, x + 2, . . . , 2.3
we obtain p ∨ hx =
x
X
n=0
pnh
n∗
x x ∈ N
1
, where the infinite summation in 2.2 has been reduced to a finite summation. In particular we have that p ∨h0 =
p0 0. Thus, p ∨ h ∈ P
10
. Sundt 1992 showed that if p is R
k
[a, b], then p ∨ hx =
x
X
y=1
p ∨ hx − y
k
X
i=1
ai + bi
i y
x h
i∗
y x ∈ N
1+
, 2.4
i.e. p ∨ h is R
k
[c, d] with cy =
k
X
i=1
aih
i∗
y, dy = y
k
X
i=1
bi i
h
i∗
y y ∈ N
1+
. 2.5
B. Sundt Insurance: Mathematics and Economics 27 2000 123–136 125
2C. From 2.1 we see that the distribution R
∞
[0, b] satisfies the recursion pn =
1 n
n
X
i=1
bipn − i n ∈ N
1+
, and by solving for bn we obtain
bn = 1
p0 npn −
n−1
X
i=1
bipn − i n ∈ N
1+
. Thus, for any p ∈ R
∞
there exists a unique function b such that p can be represented as R
∞
[0, b]. We call this b the De Pril transform of p and denote it by ϕ
p
. For convenience we extend the definition of the De Pril transform with ϕ
p
0 = 0. We now have pn =
1 n
n
X
i=1
ϕ
p
ipn − i n ∈ N
1+
2.6 ϕ
p
n = 1
p0 npn −
n−1
X
i=1
ϕ
p
ipn − i n ∈ N
1
, 2.7
making the convention that P
s i=r
= 0 when s r. From 2.5 we see that if p ∈ P
10
and h ∈ P
1+
, then ϕ
p∨h
y = y
x
X
i=1
ϕ
p
i i
h
i∗
y y ∈ N
1+
, 2.8
which was given by Sundt 1995. He also showed that for f
1
, f
2
, . . . , f
r
∈ P
10
ϕ
∗
r j =1
f
j
=
r
X
j =1
ϕ
f
j
. 2.9
Let p be R
k
[a, b]. Sundt 1995 showed that then ϕ
p
n = nan + bn +
k
X
i=1
aiϕ
p
n − i n ∈ N
1+
, 2.10
with an = bn = 0 for all n k and ϕ
p
n = 0 for all negative n. From this and 2.5 we obtain that ϕ
p∨h
x = x
k
X
y=1
ay + by
y h
y∗
x +
x−1
X
y=1
ϕ
p∨h
x − y
k
X
z=1
azh
z∗
y x ∈ N
1
. 2.11
This recursion is discussed by Sundt and Ekuma 1999. Sundt 1995 showed that a distribution in P
10
has a non-negative De Pril transform if and only if it is infinitely divisible, or, equivalently, that it can be represented as a compound Poisson distribution.
2D. Formulae 2.7, 2.9 and 2.6 can be applied for numerical evaluation of convolutions of distributions in P
10
. For each of the distributions we can evaluate its De Pril transform by the recursion 2.7. Then we use 2.9 to find the De Pril transform of the convolution, and finally we apply this De Pril transform in 2.6 for recursive
evaluation of the convolution. In practice, the individual distributions in the convolutions will often be represented as compound distributions. In that case, their De Pril transforms can be evaluated by 2.8.
126 B. Sundt Insurance: Mathematics and Economics 27 2000 123–136
Unfortunately, for large x, numerical evaluation of 2.8 can be rather time-consuming. Therefore, methods have been developed under which the De Pril transform of the counting distribution is replaced with a function that is equal
to zero for all values of the argument larger than some integer r. Such approximations were first discussed within the terminology of De Pril transforms by Dhaene and Sundt 1998. Of earlier references, we mention De Pril 1989
and Dhaene and De Pril 1994. With such approximations to De Pril transforms, the resulting approximations to distributions are not necessarily distributions themselves. Dhaene and Sundt 1998 therefore extended the definition
2.7 of the De Pril transform to functions in F
10
, the class of functions on N
1
with a positive mass at zero, and they discussed properties of the De Pril transform within this framework. In particular they showed that the additivity
property 2.9 still holds for functions in F
10
, and that 2.8 holds when p ∈ F
10
and h ∈ F
1+
, being the class of functions on N
1+
. From 2.6 we see that the De Pril transform of a function in F
10
determines the function only up to a multiplicative constant. However, a distribution in P
10
is uniquely determined by its De Pril transform as it should sum to one. As the recursion 2.1 determines a function p ∈ F
10
only up to a multiplicative constant, Dhaene and Sundt 1998 defined p to be in the form R
k
[a, b] if it satisfies that recursion, and showed that 2.10 still holds for such functions. Extension of 2.11 to the case when p ∈ F
10
is in the form R
k
[a, b] and h ∈ F
1+
, is trivial. 2E. Recursions like 2.4 were originally developed for probability functions. Dhaene et al. 1999 discuss how
one can deduce, from recursions for probability functions, recursions for cumulations like cumulative distribution functions and functions of even higher order. In this connection they defined for functions f ∈ F
10
the cumulation operator Ŵ by
Ŵf x =
x
X
y=0
f y x ∈ N
1
. We see that we have Ŵf = f ∗ u, where u ∈ F
10
is defined by ux = 1
x ∈ N
1
. 2.12
From 2.7 we obtain that ϕ
u
x = 1 x ∈ N
1+
, 2.13
and application of 2.9 gives ϕ
Ŵ
t
f
x = ϕ
f
x + t x ∈ N
1+
; t ∈ N
1
, 2.14
which was proved by Dhaene et al. 1999.
3. Multivariate De Pril transform
