Positive recurrent Markov chain
$begingroup$
Is it true that for a recurrent Markov chain on a countable state space $S$, if there exist a state $xin S$ such that
$$ lim_{ntoinfty} p_n(x, y) > 0, $$
then for all $zin S$,
$$ lim_{ntoinfty} p_n(x, y) > 0? $$
If so, could the assumption of recurrent chain be weakened to reducible?
markov-chains
asked Dec 26 '18 at 18:39
p-valuep-value
224111
$endgroup$
add a comment |
$begingroup$
Is it true that for a recurrent Markov chain on a countable state space $S$, if there exist a state $xin S$ such that
$$ lim_{ntoinfty} p_n(x, y) > 0, $$
then for all $zin S$,
$$ lim_{ntoinfty} p_n(x, y) > 0? $$
If so, could the assumption of recurrent chain be weakened to reducible?
markov-chains
asked Dec 26 '18 at 18:39
p-valuep-value
224111
$endgroup$
$begingroup$
Simply use $$p_n(z,y)geqslant p_k(z,x)p_{n-k}(x,y)$$ for every $n>k$, for a suitable, well-chosen, value of $k$. Recurrence is offtopic. Irreducibility is (much!) needed.
$endgroup$
– Did
Dec 31 '18 at 20:14
add a comment |
$begingroup$
Is it true that for a recurrent Markov chain on a countable state space $S$, if there exist a state $xin S$ such that
$$ lim_{ntoinfty} p_n(x, y) > 0, $$
then for all $zin S$,
$$ lim_{ntoinfty} p_n(x, y) > 0? $$
If so, could the assumption of recurrent chain be weakened to reducible?
markov-chains
asked Dec 26 '18 at 18:39
p-valuep-value
224111
$endgroup$
Is it true that for a recurrent Markov chain on a countable state space $S$, if there exist a state $xin S$ such that
$$ lim_{ntoinfty} p_n(x, y) > 0, $$
then for all $zin S$,
$$ lim_{ntoinfty} p_n(x, y) > 0? $$
If so, could the assumption of recurrent chain be weakened to reducible?
markov-chains
markov-chains
asked Dec 26 '18 at 18:39
p-valuep-value
224111
asked Dec 26 '18 at 18:39
p-valuep-value
224111
asked Dec 26 '18 at 18:39
p-valuep-value
224111
asked Dec 26 '18 at 18:39
p-valuep-value
224111
asked Dec 26 '18 at 18:39
p-valuep-value
224111
224111
$begingroup$
Simply use $$p_n(z,y)geqslant p_k(z,x)p_{n-k}(x,y)$$ for every $n>k$, for a suitable, well-chosen, value of $k$. Recurrence is offtopic. Irreducibility is (much!) needed.
$endgroup$
– Did
Dec 31 '18 at 20:14
add a comment |
$begingroup$
Simply use $$p_n(z,y)geqslant p_k(z,x)p_{n-k}(x,y)$$ for every $n>k$, for a suitable, well-chosen, value of $k$. Recurrence is offtopic. Irreducibility is (much!) needed.
$endgroup$
– Did
Dec 31 '18 at 20:14
$begingroup$
Simply use $$p_n(z,y)geqslant p_k(z,x)p_{n-k}(x,y)$$ for every $n>k$, for a suitable, well-chosen, value of $k$. Recurrence is offtopic. Irreducibility is (much!) needed.
$endgroup$
– Did
Dec 31 '18 at 20:14
$begingroup$
Simply use $$p_n(z,y)geqslant p_k(z,x)p_{n-k}(x,y)$$ for every $n>k$, for a suitable, well-chosen, value of $k$. Recurrence is offtopic. Irreducibility is (much!) needed.
$endgroup$
– Did
Dec 31 '18 at 20:14
add a comment |
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$begingroup$
Simply use $$p_n(z,y)geqslant p_k(z,x)p_{n-k}(x,y)$$ for every $n>k$, for a suitable, well-chosen, value of $k$. Recurrence is offtopic. Irreducibility is (much!) needed.
$endgroup$
– Did
Dec 31 '18 at 20:14