Stochastic Monotonicity: Dependence on the Initial State Distribution

Abstract

In the first three sections of this chapter we consider a natural birth-death process for which 0 is a reflecting barrier with initial distribution vector q = (q₀, q₁,.....)^T, where q_n = δ_in for some fixed i and all n. The process is denoted by {X_i(t)} = {X_i (t): 0 ≤ t < ∞} and the corresponding time dependent vector defined by (4.1.7), which, according to (4.1.10), is the ith row of the matrix E(t) = (e_ij(t)), is denoted by e_i (t), i.e.,

$$ \underline e \left( t \right) = {\left( {{e_{i0}}\left( t \right),{\kern 1pt} {e_{i1}}\left( t \right),......} \right)^T} $$

.