Distinguish positive recurrent, null recurrent and transient
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Given transition probability as below, how can I tell that the Markov Chain $X_n$ is a positive recurrent, null recurrent or transient?
$$p(x,0)=1/(x^2+1),quad p(x,x+1)=(x^2+1)/(x^2+2)$$ state space $S={0,1,2,...}$
I have tried to calculate $alpha(x)$ for $x in S$, but it turns out to be $1$ for all $x$. So it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
probability markov-chains
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
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add a comment |
$begingroup$
Given transition probability as below, how can I tell that the Markov Chain $X_n$ is a positive recurrent, null recurrent or transient?
$$p(x,0)=1/(x^2+1),quad p(x,x+1)=(x^2+1)/(x^2+2)$$ state space $S={0,1,2,...}$
I have tried to calculate $alpha(x)$ for $x in S$, but it turns out to be $1$ for all $x$. So it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
probability markov-chains
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
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Hint: Can you compute $P_0(T_0 text{finite})$?
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– Did
Oct 8 '15 at 19:47
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More details about it?
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– MichaelJ
Oct 8 '15 at 19:56
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Sure, you first.
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– Did
Oct 8 '15 at 19:57
$begingroup$
I have tried to calculate alpha(x) for x in S, but it turns out to be 1 for all x. so it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
$endgroup$
– MichaelJ
Oct 8 '15 at 20:46
2
$begingroup$
Please recheck the transition probabilities: why don't they add up to 1?
$endgroup$
– user147263
Oct 9 '15 at 1:06
add a comment |
$begingroup$
Given transition probability as below, how can I tell that the Markov Chain $X_n$ is a positive recurrent, null recurrent or transient?
$$p(x,0)=1/(x^2+1),quad p(x,x+1)=(x^2+1)/(x^2+2)$$ state space $S={0,1,2,...}$
I have tried to calculate $alpha(x)$ for $x in S$, but it turns out to be $1$ for all $x$. So it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
probability markov-chains
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
$endgroup$
Given transition probability as below, how can I tell that the Markov Chain $X_n$ is a positive recurrent, null recurrent or transient?
$$p(x,0)=1/(x^2+1),quad p(x,x+1)=(x^2+1)/(x^2+2)$$ state space $S={0,1,2,...}$
I have tried to calculate $alpha(x)$ for $x in S$, but it turns out to be $1$ for all $x$. So it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
probability markov-chains
probability markov-chains
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
edited Oct 9 '15 at 1:06
user147263
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
asked Oct 8 '15 at 19:33
MichaelJMichaelJ
62
62
$begingroup$
Hint: Can you compute $P_0(T_0 text{finite})$?
$endgroup$
– Did
Oct 8 '15 at 19:47
$begingroup$
More details about it?
$endgroup$
– MichaelJ
Oct 8 '15 at 19:56
$begingroup$
Sure, you first.
$endgroup$
– Did
Oct 8 '15 at 19:57
$begingroup$
I have tried to calculate alpha(x) for x in S, but it turns out to be 1 for all x. so it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
$endgroup$
– MichaelJ
Oct 8 '15 at 20:46
2
$begingroup$
Please recheck the transition probabilities: why don't they add up to 1?
$endgroup$
– user147263
Oct 9 '15 at 1:06
add a comment |
$begingroup$
Hint: Can you compute $P_0(T_0 text{finite})$?
$endgroup$
– Did
Oct 8 '15 at 19:47
$begingroup$
More details about it?
$endgroup$
– MichaelJ
Oct 8 '15 at 19:56
$begingroup$
Sure, you first.
$endgroup$
– Did
Oct 8 '15 at 19:57
$begingroup$
I have tried to calculate alpha(x) for x in S, but it turns out to be 1 for all x. so it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
$endgroup$
– MichaelJ
Oct 8 '15 at 20:46
2
$begingroup$
Please recheck the transition probabilities: why don't they add up to 1?
$endgroup$
– user147263
Oct 9 '15 at 1:06
$begingroup$
Hint: Can you compute $P_0(T_0 text{finite})$?
$endgroup$
– Did
Oct 8 '15 at 19:47
$begingroup$
Hint: Can you compute $P_0(T_0 text{finite})$?
$endgroup$
– Did
Oct 8 '15 at 19:47
$begingroup$
More details about it?
$endgroup$
– MichaelJ
Oct 8 '15 at 19:56
$begingroup$
More details about it?
$endgroup$
– MichaelJ
Oct 8 '15 at 19:56
$begingroup$
Sure, you first.
$endgroup$
– Did
Oct 8 '15 at 19:57
$begingroup$
Sure, you first.
$endgroup$
– Did
Oct 8 '15 at 19:57
$begingroup$
I have tried to calculate alpha(x) for x in S, but it turns out to be 1 for all x. so it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
$endgroup$
– MichaelJ
Oct 8 '15 at 20:46
$begingroup$
I have tried to calculate alpha(x) for x in S, but it turns out to be 1 for all x. so it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
$endgroup$
– MichaelJ
Oct 8 '15 at 20:46
2
2
$begingroup$
Please recheck the transition probabilities: why don't they add up to 1?
$endgroup$
– user147263
Oct 9 '15 at 1:06
$begingroup$
Please recheck the transition probabilities: why don't they add up to 1?
$endgroup$
– user147263
Oct 9 '15 at 1:06
add a comment |
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$begingroup$
Hint: Can you compute $P_0(T_0 text{finite})$?
$endgroup$
– Did
Oct 8 '15 at 19:47
$begingroup$
More details about it?
$endgroup$
– MichaelJ
Oct 8 '15 at 19:56
$begingroup$
Sure, you first.
$endgroup$
– Did
Oct 8 '15 at 19:57
$begingroup$
I have tried to calculate alpha(x) for x in S, but it turns out to be 1 for all x. so it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.
$endgroup$
– MichaelJ
Oct 8 '15 at 20:46
2
$begingroup$
Please recheck the transition probabilities: why don't they add up to 1?
$endgroup$
– user147263
Oct 9 '15 at 1:06