Section 15.2

Long-Run Behaviour

Stationary distributions, doubly stochastic chains, reflecting barriers, limits of noisy relays.

Toolkit

For an ergodic chain, the stationary distribution $\pi$ solves $\pi P = \pi$ , $\sum_i \pi_i = 1$ , and $\pi_i = \lim_{n\to\infty}\mathbb{P}(X_n = i)$ regardless of the start.

If $P$ is doubly stochastic (rows and columns both sum to $1$ ) on a finite state space of size $N$ , then $\pi_i = 1/N$ for all $i$ , since the uniform vector is preserved: $\sum_i \frac1N P_{ij} = \frac1N$ .
Birth-death chains on $\{0,1,\dots\}$ with up-probability $p_i$ and down-probability $q_i$ satisfy detailed balance $\pi_i p_i = \pi_{i+1} q_{i+1}$ , giving $\pi_{i+1} = \pi_i \cdot p_i/q_{i+1}$ ; a constant ratio $p/q<1$ yields a geometric stationary law, truncated or reflected as the barrier dictates.
For two-state chains, balancing flow $A\to B$ against flow $B\to A$ gives $\pi_A p_{AB} = \pi_B p_{BA}$ .
A linear recursion $x_{n+1} = a x_n + b$ with $|a|<1$ converges to the fixed point $x^\ast = b/(1-a)$ , independent of $x_0$ ; this governs the limiting bias of repeated noisy transmissions.

Problems