$R$-Theory for Markov Chains on a General State Space I: Solidarity Properties and $R$-Recurrent Chains

Abstract

This paper develops, for a Markov chain $\{X_n\}$ on a general space $(\mathscr{X}, \mathscr{F})$ with $n$-step transition probabilities $P^n(x, A), x \in \mathscr{X}, A \in \mathscr{F}$, a theory analogous to that of Vere-Jones for Markov chains on the integers. If the chain is $\phi$-irreducible there is a partition $\mathscr{K}$ of $\mathscr{X}$ such that $\phi$-almost all of the power series $G_z(x, A) = \sum_n P^n(x, A)z^n$ have a common radius of convergence $R$ for $A$ in any element of $\mathscr{K}$, and they all diverge ($R$-recurrence) or all converge ($R$-transience) for $z = R$. The $R$-recurrent case is then investigated, and it is shown that there exist essentially unique non-zero solutions $Q, f$ to the $R$-subinvariant equations $Q \geqq RQP$ and $f \geqq RPf$, and that $Q$ and $f$ satisfy these inequalities with equality: a relationship between $Q$ and $f$ and first-entrance probabilities is also established. Further, if $\{X_n\}$ is aperiodic, $\lim_{n\rightarrow\infty} R^nP^n(x, A) = f(x)Q(A)/\int_\mathscr{X} f(y)Q(dy)$ for almost all $x \in \mathscr{X}$ and $A$ in any element of a second partition. The methods used are probabilistic and depend mainly on generating function techniques: it is pointed out that these techniques do not depend on the substochasticity of the transition probabilities, and hence the results are true in a much wider context.