Transient or Recurrent behavior of Random walk on positive integer
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I know that the random walk on Z is transient or recurrent depending upon the probability. Now I think that the random walk on Z+ is also null recurrent or non-null recurrent or transient(probability of transition from i to i+1 is p and i+1 to i is q and from 0 to 0 is q, 0 to 1 is p where p+q=1, i>=0) depending on p. But I can't approach for the proper proof. please Help me. Thanks in advance.
probability-theory statistics markov-chains
asked Nov 19 at 9:11
Kousik Das
11
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up vote
-1
down vote
favorite
I know that the random walk on Z is transient or recurrent depending upon the probability. Now I think that the random walk on Z+ is also null recurrent or non-null recurrent or transient(probability of transition from i to i+1 is p and i+1 to i is q and from 0 to 0 is q, 0 to 1 is p where p+q=1, i>=0) depending on p. But I can't approach for the proper proof. please Help me. Thanks in advance.
probability-theory statistics markov-chains
asked Nov 19 at 9:11
Kousik Das
11
It should be obvious that if $p < q$ then it is recurrent. As to how to approach it, consider the "first step analysis" of the escape probabilities.
– Will M.
Nov 20 at 6:47
Every time my intuition says if p is not equal to 1 then for any state i it is recurrent, it can't be transient. If p=1 then only it is transient. But It may be wrong. Please help me
– Kousik Das
Nov 20 at 19:07
Actually how to prove for 0.5<p<=1, sum over f_ii(n)(first entrance probability in n step) from 1 to infinity is less than 1 to find i as transient state?
– Kousik Das
Nov 20 at 19:16
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I know that the random walk on Z is transient or recurrent depending upon the probability. Now I think that the random walk on Z+ is also null recurrent or non-null recurrent or transient(probability of transition from i to i+1 is p and i+1 to i is q and from 0 to 0 is q, 0 to 1 is p where p+q=1, i>=0) depending on p. But I can't approach for the proper proof. please Help me. Thanks in advance.
probability-theory statistics markov-chains
asked Nov 19 at 9:11
Kousik Das
11
I know that the random walk on Z is transient or recurrent depending upon the probability. Now I think that the random walk on Z+ is also null recurrent or non-null recurrent or transient(probability of transition from i to i+1 is p and i+1 to i is q and from 0 to 0 is q, 0 to 1 is p where p+q=1, i>=0) depending on p. But I can't approach for the proper proof. please Help me. Thanks in advance.
probability-theory statistics markov-chains
probability-theory statistics markov-chains
asked Nov 19 at 9:11
Kousik Das
11
asked Nov 19 at 9:11
Kousik Das
11
edited Nov 20 at 6:14
asked Nov 19 at 9:11
Kousik Das
11
asked Nov 19 at 9:11
Kousik Das
11
asked Nov 19 at 9:11
Kousik Das
11
11
It should be obvious that if $p < q$ then it is recurrent. As to how to approach it, consider the "first step analysis" of the escape probabilities.
– Will M.
Nov 20 at 6:47
Every time my intuition says if p is not equal to 1 then for any state i it is recurrent, it can't be transient. If p=1 then only it is transient. But It may be wrong. Please help me
– Kousik Das
Nov 20 at 19:07
Actually how to prove for 0.5<p<=1, sum over f_ii(n)(first entrance probability in n step) from 1 to infinity is less than 1 to find i as transient state?
– Kousik Das
Nov 20 at 19:16
add a comment |
It should be obvious that if $p < q$ then it is recurrent. As to how to approach it, consider the "first step analysis" of the escape probabilities.
– Will M.
Nov 20 at 6:47
Every time my intuition says if p is not equal to 1 then for any state i it is recurrent, it can't be transient. If p=1 then only it is transient. But It may be wrong. Please help me
– Kousik Das
Nov 20 at 19:07
Actually how to prove for 0.5<p<=1, sum over f_ii(n)(first entrance probability in n step) from 1 to infinity is less than 1 to find i as transient state?
– Kousik Das
Nov 20 at 19:16
It should be obvious that if $p < q$ then it is recurrent. As to how to approach it, consider the "first step analysis" of the escape probabilities.
– Will M.
Nov 20 at 6:47
It should be obvious that if $p < q$ then it is recurrent. As to how to approach it, consider the "first step analysis" of the escape probabilities.
– Will M.
Nov 20 at 6:47
Every time my intuition says if p is not equal to 1 then for any state i it is recurrent, it can't be transient. If p=1 then only it is transient. But It may be wrong. Please help me
– Kousik Das
Nov 20 at 19:07
Every time my intuition says if p is not equal to 1 then for any state i it is recurrent, it can't be transient. If p=1 then only it is transient. But It may be wrong. Please help me
– Kousik Das
Nov 20 at 19:07
Actually how to prove for 0.5<p<=1, sum over f_ii(n)(first entrance probability in n step) from 1 to infinity is less than 1 to find i as transient state?
– Kousik Das
Nov 20 at 19:16
Actually how to prove for 0.5<p<=1, sum over f_ii(n)(first entrance probability in n step) from 1 to infinity is less than 1 to find i as transient state?
– Kousik Das
Nov 20 at 19:16
add a comment |
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It should be obvious that if $p < q$ then it is recurrent. As to how to approach it, consider the "first step analysis" of the escape probabilities.
– Will M.
Nov 20 at 6:47
Every time my intuition says if p is not equal to 1 then for any state i it is recurrent, it can't be transient. If p=1 then only it is transient. But It may be wrong. Please help me
– Kousik Das
Nov 20 at 19:07
Actually how to prove for 0.5<p<=1, sum over f_ii(n)(first entrance probability in n step) from 1 to infinity is less than 1 to find i as transient state?
– Kousik Das
Nov 20 at 19:16