Natural Birth-Death Processes

Abstract

Let {λ_n,μ_n} be a set of birth-death parameters, A the associated generator (1.4.2) and {X(t): 0 ≤ t < ∞} a birth-death process with generator A. In this and the following chapters we shall be concerned with natural birth-death processes only, i.e., A is assumed to satisfy the conditions C(A) and D(A). The state -1 will be disregarded and the term transition matrix will be used for the matrix P(⋅) = (p_ij(⋅)), where i, j = 0, 1,....... Since the properties to be discussed in this chapter are independent of the initial distribution of the process, we shall often identify the birth-death process {X(t)} with its transition matrix P(.). KARLIN and McGREGOR (1957^a, 1959^b) have proved the following important feature of the transition matrix P(⋅) of a natural birth-death process: (2.1.1) P(t) is strictly totally positive (STP) for t > 0, which means that every subdeterminant of P(t) is strictly positive for t > 0. KARLIN and McGREGOR (1959^a) showed that property (2.1.1) is characteristic for birth-death processes in the class of Markov processes as defined in section 1.1. An immediate conclusion from (2.1.1) is (2.1.2) p_ij(t) > 0 i, j = 0, 1,...; t > 0.