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In a positive recurrent birth-death-immigration jump chain, let $I_n$ and $A_n$ count immigration events and upward steps before time $n$. The centered immigration indicators form bounded martingale differences; the strong law for martingales with bounded increments and the countable-state Markov chain ergodic theorem give $I_n/n\to {\sum }_k{\pi }_k\theta /(k+\theta )$ almost surely. The analogous argument for upward steps gives $A_n/n\to {\sum }_k{\pi }_k{u}_k$. Summing detailed balance gives ${\sum }_k{\pi }_k{u}_k={\sum }_k{\pi }_k{d}_k=1/2$, since every step is upward or downward. Therefore $I_n/A_n$ converges to the displayed fraction. For this chain the explicit stationary distribution reduces the fraction to $1-p/(1-p)$.

Under the specified evolution rule, conditional on the entire history, the next event's probability distribution depends only on the current population size. This is the time-homogeneous Markov property. Birth and immigration both increase the size by one, while death decreases it by one. Writing $q=1-p$, the transition probabilities on ${\mathbb{Z}}_{\ge 0}$ areand all other entries are zero. In particular $P(0,1)=1$ and $d_0=0$. The row sum is $[kp+\theta +kq]/(k+\theta )=1$. Every neighboring pair communicates, so this is an irreducible Markov chain, specifically a birth-death-immigration jump chain.