Stochastic Monotonicity: General Results

Abstract

Let {λ_n, μ_n: n = 0, 1......}, with μ₀ = 0, be the set of parameters of a natural birth-death process {X(t) : 0 ≤ t < ∞} where 0 is a reflecting barrier. The initial distribution vector of {X(t)} will be denoted by q = (q₀, q₁,....)^T, i.e.,

$${q_{i}} = {p_{i}}\left( 0 \right) = \Pr \left\{ {X\left( 0 \right) = i} \right\}$$

((1))

otherwise the notation of section 1.4 will be used. We have

$$\underline {q \geqslant \underline 0 ,\underline {{q^{T}}\underline 1 } } = 1,$$

((4.1.1) )

where vector inequality is defined by (1.2.15) and 0 and 1 are the column vectors consisting of 0’s and 1’s, respectively. We recall that

$$\underline {{P^{T}}\left( t \right)P\left( s \right)}$$

((4.1.2))

where p^T(t) = (p₀(t), p₁(t),.....) with p_i(t) = Pr{X(t) = i} and P(t) the transi­tion matrix of {X(t)}. More generally one has

$$\underline {{P^{T}}} \left( t \right)\underline 1 = 1,$$

((4.1.3))

From (1.4.14) and (4.1.1) it follows that for all t ≥ 0

$$\underline {{P^{T}}\left( t \right)} \underline 1 = 1,$$

((4.1.4))

so that p(t) is indeed a probability distribution vector.