A test of uniform ergodicity and strong stability for Markov chains with general phase space. (Russian) Zbl 0562.60070

Kartashov, N. V.

Teor. Veroyatn. Mat. Stat. 30, 65-81 (1984).

Let (E,\({\mathcal E})\) be a general state space. Denote by m\({\mathcal E}\) (resp. f\({\mathcal E})\) the set of all finite measures on \({\mathcal E}\) (\({\mathcal E}\)- measurable functions). For a given kernel Q(x,A) on (E,\({\mathcal E})\), one can define two linear transforms in the usual way: \[ \mu Q(\cdot)=\int \mu (dx)Q(x,\cdot),\quad \mu \in m{\mathcal E}; \]
\[ Qf(\cdot)=\int Q(\cdot,dy)f(y),\quad f\in f{\mathcal E}. \] The kernel corresponding to an \(f\in f{\mathcal E}\) and a \(\mu\in m{\mathcal E}\) is defined by \(f\circ \mu (x,A)=f(x)\mu (A).\) Similarly, we have \(\mu f=\int \mu (dx)f(x)\). Given a norm \(\| \cdot \|\) on \(m{\mathcal E}\), set \({\mathcal M}=\{\mu \in m{\mathcal E}:\) \(\| \mu \| <\infty \}\), \(\| g\| =\sup \{| \mu g|:\) \(\| \mu \| \leq 1\}\) and \({\mathcal N}=\{g:\) \(\| g\| <\infty \}\) which gives the equivalence classes: \(g=\{f\in f{\mathcal E}:\) \(\mu f=\mu g\), \(\mu\in {\mathcal M}\}\). Finally the norm of a kernel Q is given by \(\| Q\| =\sup (\| \mu Q\|:\) \(\| \mu \| \leq 1).\)
Now, let \(X=(X_ t:\) \(t=0,1,...)\) be a Markov chain with state space (E,\({\mathcal E})\) and transition probability kernel P(x,A). Consider a quite general norm \(\| \cdot \|\) on m\({\mathcal E}\), such that \(\| P\| <\infty\). With respect to the norm, the chain X is called uniform ergodic if it has a unique stationary distribution \(\pi\) and \(\| t^{- 1}\sum^{t-1}_{s=0}P^ s-1\circ \pi \| \to 0,\) \(t\to \infty\); the chain is called strongly stable if there exists a kernel Q in each neighborhood \(\{\) \(Q: \| Q-P\| <\epsilon \}\), so that Q has a unique stationary distribution \(\nu\) and \(\| \nu -\pi \| \to 0\), as \(\| Q-P\| \to 0.\)
The author proves in the paper that the two concepts are indeed equivalent. For this, some necessary and sufficient conditions are presented.

Reviewer: M.Chen

MSC:

60J10

Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

Markov chain with general state space; uniform ergodic; unique stationary distribution; strongly stable