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edit approved Feb 21 at 11:51
The Amplitwist
edit approved Feb 21 at 11:51
The Amplitwist
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I believe that it is not hard to show that $(P^n)_{i,null} \to 1$ as $n\to \infty$ which responds your question if I understand well.

The proof follows from the well known Borel-Cantelli LemmaBorel-Cantelli Lemma. Notice that the probability of not arriving to the null state is in each step smaller than one (and far from one since there are finitely many states), so you get that the probability of not reaching the null state in a given step is smaller than $\lambda^n$ where $n$ is the number of steps. This implies that the probability of never reaching the null state is zero by the Lemma above.

I believe that it is not hard to show that $(P^n)_{i,null} \to 1$ as $n\to \infty$ which responds your question if I understand well.

The proof follows from the well known Borel-Cantelli Lemma. Notice that the probability of not arriving to the null state is in each step smaller than one (and far from one since there are finitely many states), so you get that the probability of not reaching the null state in a given step is smaller than $\lambda^n$ where $n$ is the number of steps. This implies that the probability of never reaching the null state is zero by the Lemma above.

I believe that it is not hard to show that $(P^n)_{i,null} \to 1$ as $n\to \infty$ which responds your question if I understand well.

The proof follows from the well known Borel-Cantelli Lemma. Notice that the probability of not arriving to the null state is in each step smaller than one (and far from one since there are finitely many states), so you get that the probability of not reaching the null state in a given step is smaller than $\lambda^n$ where $n$ is the number of steps. This implies that the probability of never reaching the null state is zero by the Lemma above.

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rpotrie
rpotrie
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I believe that it is not hard to show that $(P^n)_{i,null} \to 1$ as $n\to \infty$ which responds your question if I understand well.

The proof follows from the well known Borel-Cantelli Lemma. Notice that the probability of not arriving to the null state is in each step smaller than one (and far from one since there are finitely many states), so you get that the probability of not reaching the null state in a given step is smaller than $\lambda^n$ where $n$ is the number of steps. This implies that the probability of never reaching the null state is zero by the Lemma above.