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recently I was reading the article

Asymptotics of a class of Markov processes which are not in general irreducible. Ann. Probab. 16, 3 (1988), 1333–1347.

by Bhattacharya, R. N., and Lee, O.

The authors consider the $\leq$ relation defined by: $$ (x_1, \ldots, x_n)\leq (a_1, \dots a_n)~\text{iff}~x_1\leq a_1, \ldots, x_n\leq a_n. $$ I was reading the proof of Theorem 3.1, and I'm stuck in the conclusion at the end of the equation (3.5). It seems to me that the authors are using the fact that

$$\{{y\leq x_{0}\}}\cup \{{y>x_{0}\}}=\mathbb{R}^n$$ is the whole space, which is not true for $n\geq 2$. As an example, for $n=2$, $\{(x,y)\leq (0,0)\}$ equals the first quadrant and $\{(x,y)> (0,0)\}$ equals the third quadrant.

Obs 1: In the original paper the relation "$\leq$" is not clearly specified, however in this Corrigendum (of another part of the paper) the authors set the relation defined as above

It seems to me that the argument of the authors works for $n=1$ where $\leq$ is a total order. However for $n\geq 2$ we have the above reasoning. Has someone already attempt for this question? Has the authors another corrigendum?

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Yes, this seems to be the authors's logic, apparently erroneous. I suggest you write to them for clarification.

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Indeed, the argument seems to be erroneous.

From the assumptions however you visit infinitely many times the set $\{y\leq x_0\}$, say, and moreover the inter-return times have exponential tails. Now, consider the chain at those time. This is again a Markov chain, and has an invariant measure, call it $\hat \pi$. At these times you can deduce an invariance principle (by the method in the paper), and then one can interpolate from these times to all $n$ due to the exponential tails mentioned above (at least, this holds for bounded $f$).

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