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

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)