= Solution
If $\vartheta>1$, the general level solution is a constant multiple of $(1,1)^T$ plus a multiple of $\vartheta^i(b,1)^T$. Boundedness of probabilities removes the growing component, and $h_{W_0}=1$ then fixes the constant to one. If $\vartheta=1$, the nonzero Jordan part produces linear growth unless the initial vector lies in the one-dimensional eigenspace; the same conclusion follows. Thus
$$
\boxed{h_{C_i}=h_{W_i}=1\quad\text{for all }i.}
$$
Starting at $W_0$, the chain leaves to $C_0$ and returns with probability one. An irreducible chain with a recurrent state has all states recurrent, so \b[the chain is recurrent] in both cases.
Back to article page
