Insurance: Mathematics and Economics 38 (2006) 391–405 The preservation of classes of discrete distributions under convolution and mixing Kristina P. Pavlova a, ∗ , Jun Cai b , Gordon E. Willmot b a Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ont., Canada N6A 5B7 b Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received February 2005; received in revised form May 2005; accepted 13 October 2005 Abstract In this paper we consider some widely utilized classes of discrete distributions and aim to provide a systematic overview about their preservation under convolution and mixing. Moreover, inclusion properties among these classes are discussed. This paper will serve as a detailed reference for the study and applications of the preservation of the classes of discrete distributions. © 2005 Elsevier B.V. All rights reserved. Keywords: Counting random variable; Discrete distribution; Convolution; Mixing; Discrete equilibrium distribution; Zero-truncated discrete distri- bution; Discrete hazard rate; Discrete failure rate; Discrete mean residual lifetime; Inclusion; Preservation 1. Introduction A counting random variable X is a non-negative random variable with support N ={0, 1,...}. The probability mass function (p.f.) f of X is given by f (n) = Pr{X = n},n = 0, 1,..., the distribution function F of X satisfies F (x) = Pr{X ≤ x}= ∑ x i=0 f (i) for all x ∈ N, and the survival function F of X satisfies F (x) = 1 - F (x) = ∑ ∞ i=x+1 f (i) for all x ∈ N. Furthermore, for all x =-1, -2,..., we have f (x) = 0,F (x) = 0, and F (x) = 1. The distribution of a counting random variable is called a discrete distribution. In particular, if f (0) = Pr{X = 0}= 0, or a counting random variable X has a support on N + ={1, 2,...}, we say that the discrete distribution is zero-truncated. Moreover, we denote N - = {-1, 0, 1,...}. In addition, we denote the discrete hazard rate or failure rate of the discrete distribution function F by h F (x) = Pr{X = x} Pr{X ≥ x} = f (x) f (x) + F (x) for x ∈ N and Pr{X ≥ x} > 0. Furthermore, we denote the discrete mean residual lifetime of F by r F (x) = E{X - x|X>x}= ∑ ∞ i=x F (i) F (x) for x ∈ N and F (x) > 0. ∗ Corresponding author. E-mail addresses: kpavlova@stats.uwo.ca (K.P. Pavlova), jcai@math.uwaterloo.ca (J. Cai), gewillmo@math.uwaterloo.ca (G.E. Willmot). 0167-6687/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2005.10.001