Stationary law of a finite birth-and-death chain

The matrix $P$ is a transition matrix: its entries are nonnegative by the hypotheses, and each row sums to one, since row $i$ with $1\le i\le m-1$ has the three entries $q_i,p_i$ and $1-p_i-q_i$, row $0$ has $p_0$ and $1-p_0$, and row $m$ has $q_m$ and $1-q_m$.

Every weight is a positive finite number: $w_0=1$ and each $w_i$ is a finite product of positive ratios $p_j\mathrm{/}{q}_{j+1}$, since all $p_j,q_{j+1}>0$; consequently $W:={\sum }_{k=0}^m{w}_k$ is a finite sum of positive terms, so $0<W<+\mathrm{\infty }$.

Detailed balance holds on every edge: for $0\le i\le m-1$ the recursion gives $w_{i+1}=w_i{p}_i\mathrm{/}{q}_{i+1}$, hence $w_i{p}_i=w_{i+1}{q}_{i+1}$.

Detailed balance holds for every pair of states: if $x,y$ are distinct and non-adjacent then $P(x,y)=P(y,x)=0$ by hypothesis, so both sides vanish; if $x=y$ then both sides equal $w_{x}P(x,x)$; and the remaining case is the adjacent pair of step 2.1.

Define $\pi (i):=w_i\mathrm{/}W$. By step 1.2 each $\pi (i)$ is a finite nonnegative number, and ${\sum }_{i=0}^m\pi (i)=W\mathrm{/}W=1$, so $\pi$ is a probability vector; moreover $\pi (x)P(x,y)=\pi (y)P(y,x)$ for all $x,y$, because step 3.1 multiplies by the common positive factor $1\mathrm{/}W$. Hence $\pi$ is a reversible probability distribution for $P$ in the sense of [F2].

By [F3] the reversible probability distribution $\pi$ satisfies $\pi P=\pi$, so it is a stationary law for the birth-and-death chain.

Boundary and scope cases: for $m=0$ the state space is $\{0\}$, the products over $i$ are empty, $w_0=1$, $\pi (0)=1$ and both the detailed-balance identity and stationarity are trivial, so the formula remains valid when the positivity hypotheses are vacuous; if an interior denominator $q_{i+1}$ vanishes, the displayed recursion is undefined. If some $p_i=0$ but all $q_{i+1}>0$, it still gives finite nonnegative weights, with $w_0=1$, and the same detailed-balance and normalization argument gives a stationary law, possibly with zero masses. Either missing directed edge destroys irreducibility on the full interval, but strict positivity of both directions is needed only for positive weights in step 1.2, not for the recursion identity when all denominators are positive; the holding probabilities $P(i,i)$ never enter the detailed-balance identities; the finite sum $W$ is legitimately inverted, and no normalization of an infinite measure is attempted, so the countable birth-and-death case requires a separate summability hypothesis and is not claimed here; and no choice principle is used, all quantities being determined by the finite data.