A question about Markov Chain.
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Suppose $X_n$ has Markov property. Show that for any $n,r in Bbb N,i in S,A subset S^n, B subset S^r$
$P[(X_n+1, cdots , X_n+r) in B mid X_n=i,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n=i]$.
If further $C subset S$ then,
$P[(X_n+1, cdots , X_n+r) in B mid X_n in C,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n in C]$.
How do I proceed using Markov property? Please help me in this regard.
Thank you very much.
probability probability-theory stochastic-processes markov-process
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up vote
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Suppose $X_n$ has Markov property. Show that for any $n,r in Bbb N,i in S,A subset S^n, B subset S^r$
$P[(X_n+1, cdots , X_n+r) in B mid X_n=i,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n=i]$.
If further $C subset S$ then,
$P[(X_n+1, cdots , X_n+r) in B mid X_n in C,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n in C]$.
How do I proceed using Markov property? Please help me in this regard.
Thank you very much.
probability probability-theory stochastic-processes markov-process
Maybe I'm wrong, but it seems there is a counterexample to the second statement given here (see the answer with 103 upvotes).
â Shalop
Aug 7 at 21:09
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose $X_n$ has Markov property. Show that for any $n,r in Bbb N,i in S,A subset S^n, B subset S^r$
$P[(X_n+1, cdots , X_n+r) in B mid X_n=i,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n=i]$.
If further $C subset S$ then,
$P[(X_n+1, cdots , X_n+r) in B mid X_n in C,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n in C]$.
How do I proceed using Markov property? Please help me in this regard.
Thank you very much.
probability probability-theory stochastic-processes markov-process
Suppose $X_n$ has Markov property. Show that for any $n,r in Bbb N,i in S,A subset S^n, B subset S^r$
$P[(X_n+1, cdots , X_n+r) in B mid X_n=i,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n=i]$.
If further $C subset S$ then,
$P[(X_n+1, cdots , X_n+r) in B mid X_n in C,(X_0,cdots,X_n-1) in A]=P[(X_n+1,cdots,X_n+r) in B mid X_n in C]$.
How do I proceed using Markov property? Please help me in this regard.
Thank you very much.
probability probability-theory stochastic-processes markov-process
edited Aug 7 at 17:51
asked Aug 7 at 17:43
Debabrata Chattopadhyay.
14312
14312
Maybe I'm wrong, but it seems there is a counterexample to the second statement given here (see the answer with 103 upvotes).
â Shalop
Aug 7 at 21:09
add a comment |Â
Maybe I'm wrong, but it seems there is a counterexample to the second statement given here (see the answer with 103 upvotes).
â Shalop
Aug 7 at 21:09
Maybe I'm wrong, but it seems there is a counterexample to the second statement given here (see the answer with 103 upvotes).
â Shalop
Aug 7 at 21:09
Maybe I'm wrong, but it seems there is a counterexample to the second statement given here (see the answer with 103 upvotes).
â Shalop
Aug 7 at 21:09
add a comment |Â
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Maybe I'm wrong, but it seems there is a counterexample to the second statement given here (see the answer with 103 upvotes).
â Shalop
Aug 7 at 21:09