Extensions of Katz-Panjer families of discrete distributions

Abstract

Let Nα, β, γ be a discrete random variable whose probability atoms {pn }n∈N satisfy f(n+1) f(n) = α + β E(U n 0 ) E(U n γ ) , n= 0, 1, ..., for some α, β ∈ R, where Uγ ⌢Uniform(γ, 1), γ ∈ (−1, 1]. When γ → 1, Uγ → U1 , the degenerate random variable with unit mass at 1, and the above iterative expression is pn+1 pn = α + β n+1 for n = k, k+1, ..., used by Katz and by Panjer (k = 0), by Sundt and Jewell and by Willmot (k = 1) and, for general k ∈ N, by Hess, Lewald and Schmidt. We investigate the case Uγ ⌢Uniform(γ, 1) with γ ∈ (−1, 1) in detail for α = 0. We then construct classes Cγ of discrete infinitely divisible randomly stopped sums such that N0, β, γ ∈ Cγ . C0 is the class of compound geometric random variables, C1 is the class of compound Poissons, and |γ1 |< γ2 ≤ 1 implies Cγ1 ⊂ Cγ2 ⊆ C1.