= Immigration fraction among upward steps
{title2=$2\sum_{k\geq0}\pi_k\theta/(k+\theta)$}
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 $A_n/n\to\sum_k\pi_ku_k$. Summing <detailed balance> gives $\sum_k\pi_ku_k=\sum_k\pi_kd_k=1/2$, 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 $1-p/(1-p)$.
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