Discussion on D.C.M. Dickson & H.R. Waters Multi-Period Aggregate Loss Distributions for a Life Portfolio

In the present discussion we point out the relation of some results in Dickson & Waters (1999) to similar results in Sundt (1998a, b).We shall need some notation. For a positive integer m let m be the set of all m × 1 vectors with positive integer-valued elements and m+ = m ~ {0}. A vector will be denoted by a bold-face letter and each of its elements by the corresponding italic with a subscript denoting the number of the elements; the subscript · denotes the sum of the elements. Let m0 be the class of probability functions on m with a positive probability at 0 and m+ the class of probability functions on m+. For j = 1,…, m we introduce the m × 1 vector ej where the jth element is one and all the other elements zero. We make the convention that summation over an empty range is equal to zero.Let g ∈ m0 be the compound probability function with counting distribution with probability function v ∈ 10 and severity distribution with probability function h ∈ m+; we shall denote this compound probability function by v V h. Sundt (1998a) showed thatwhere φv denotes the De Pril transform of v, given byMotivated by (2) Sundt (1998a) defined the De Pril transform φg of g byThis defines the De Pril transform for all probability functions in m0. Insertion of (2) in (3) givesand by solving φg(X) we obtainSundt (1998a) studies the De Pril transform defined in this way and found in particular that it is additive for convolutions.