States of Markov chain and stationary distribution
$begingroup$
Let $X$ be a Markov chain with a state space $S={{0,1,2,... }}$ and a transition matrix $P$ with given $p_{i,0}=frac{i}{i+1}$ and $p_{i,i+1}=frac{1}{i+1}$, for $i=0,1,2,...$. Find out which states are transient, null and not-null. Find stationary distribution.
I have that $p_{0,0}=0, p_{1,0}=frac{1}{2}, p_{2,0}=frac{2}{3}, p_{3,0}=frac{3}{4}$ and $p_{0,1}=1, p_{1,2}=frac{1}{2}, p_{2,3}=frac{1}{3}, p_{3,4}=frac{1}{4}$.
I think that all states are persistent, so no states are transient.
There is a theorem which says that a state is null $iff lim_{n to infty} p_{ii}(n)=0$, so state $0$ is null. I guess all the other states are non-null but I don't know how to prove it.
Also I don't know how to find stationary distribution.
Please help me with the above exercise, correct me where I'm wrong, and send any tips to the rest. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Jan 6 at 12:04
MacAbraMacAbra
269210
$endgroup$
add a comment |
$begingroup$
Let $X$ be a Markov chain with a state space $S={{0,1,2,... }}$ and a transition matrix $P$ with given $p_{i,0}=frac{i}{i+1}$ and $p_{i,i+1}=frac{1}{i+1}$, for $i=0,1,2,...$. Find out which states are transient, null and not-null. Find stationary distribution.
I have that $p_{0,0}=0, p_{1,0}=frac{1}{2}, p_{2,0}=frac{2}{3}, p_{3,0}=frac{3}{4}$ and $p_{0,1}=1, p_{1,2}=frac{1}{2}, p_{2,3}=frac{1}{3}, p_{3,4}=frac{1}{4}$.
I think that all states are persistent, so no states are transient.
There is a theorem which says that a state is null $iff lim_{n to infty} p_{ii}(n)=0$, so state $0$ is null. I guess all the other states are non-null but I don't know how to prove it.
Also I don't know how to find stationary distribution.
Please help me with the above exercise, correct me where I'm wrong, and send any tips to the rest. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Jan 6 at 12:04
MacAbraMacAbra
269210
$endgroup$
$begingroup$
"so state 0 is null" How do you know? Did you prove that $limlimits_{n to infty} p_{00}(n)=0$? How?
$endgroup$
– Did
Jan 6 at 12:46
$begingroup$
I just assumed that if $p_{0,0}=0$, then $lim_{n to infty}p_{0,0}(n)=0$, I know it might be naive, but I'm not aware how to approach this problem.
$endgroup$
– MacAbra
Jan 6 at 13:10
$begingroup$
Not especially naive, but squarely wrong because obviously absurd, right? Say, do you have any serious attempt to present?
$endgroup$
– Did
Jan 6 at 13:21
add a comment |
$begingroup$
Let $X$ be a Markov chain with a state space $S={{0,1,2,... }}$ and a transition matrix $P$ with given $p_{i,0}=frac{i}{i+1}$ and $p_{i,i+1}=frac{1}{i+1}$, for $i=0,1,2,...$. Find out which states are transient, null and not-null. Find stationary distribution.
I have that $p_{0,0}=0, p_{1,0}=frac{1}{2}, p_{2,0}=frac{2}{3}, p_{3,0}=frac{3}{4}$ and $p_{0,1}=1, p_{1,2}=frac{1}{2}, p_{2,3}=frac{1}{3}, p_{3,4}=frac{1}{4}$.
I think that all states are persistent, so no states are transient.
There is a theorem which says that a state is null $iff lim_{n to infty} p_{ii}(n)=0$, so state $0$ is null. I guess all the other states are non-null but I don't know how to prove it.
Also I don't know how to find stationary distribution.
Please help me with the above exercise, correct me where I'm wrong, and send any tips to the rest. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Jan 6 at 12:04
MacAbraMacAbra
269210
$endgroup$
Let $X$ be a Markov chain with a state space $S={{0,1,2,... }}$ and a transition matrix $P$ with given $p_{i,0}=frac{i}{i+1}$ and $p_{i,i+1}=frac{1}{i+1}$, for $i=0,1,2,...$. Find out which states are transient, null and not-null. Find stationary distribution.
I have that $p_{0,0}=0, p_{1,0}=frac{1}{2}, p_{2,0}=frac{2}{3}, p_{3,0}=frac{3}{4}$ and $p_{0,1}=1, p_{1,2}=frac{1}{2}, p_{2,3}=frac{1}{3}, p_{3,4}=frac{1}{4}$.
I think that all states are persistent, so no states are transient.
There is a theorem which says that a state is null $iff lim_{n to infty} p_{ii}(n)=0$, so state $0$ is null. I guess all the other states are non-null but I don't know how to prove it.
Also I don't know how to find stationary distribution.
Please help me with the above exercise, correct me where I'm wrong, and send any tips to the rest. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
probability-theory stochastic-processes markov-chains
asked Jan 6 at 12:04
MacAbraMacAbra
269210
asked Jan 6 at 12:04
MacAbraMacAbra
269210
asked Jan 6 at 12:04
MacAbraMacAbra
269210
asked Jan 6 at 12:04
MacAbraMacAbra
269210
asked Jan 6 at 12:04
MacAbraMacAbra
269210
269210
$begingroup$
"so state 0 is null" How do you know? Did you prove that $limlimits_{n to infty} p_{00}(n)=0$? How?
$endgroup$
– Did
Jan 6 at 12:46
$begingroup$
I just assumed that if $p_{0,0}=0$, then $lim_{n to infty}p_{0,0}(n)=0$, I know it might be naive, but I'm not aware how to approach this problem.
$endgroup$
– MacAbra
Jan 6 at 13:10
$begingroup$
Not especially naive, but squarely wrong because obviously absurd, right? Say, do you have any serious attempt to present?
$endgroup$
– Did
Jan 6 at 13:21
add a comment |
$begingroup$
"so state 0 is null" How do you know? Did you prove that $limlimits_{n to infty} p_{00}(n)=0$? How?
$endgroup$
– Did
Jan 6 at 12:46
$begingroup$
I just assumed that if $p_{0,0}=0$, then $lim_{n to infty}p_{0,0}(n)=0$, I know it might be naive, but I'm not aware how to approach this problem.
$endgroup$
– MacAbra
Jan 6 at 13:10
$begingroup$
Not especially naive, but squarely wrong because obviously absurd, right? Say, do you have any serious attempt to present?
$endgroup$
– Did
Jan 6 at 13:21
$begingroup$
"so state 0 is null" How do you know? Did you prove that $limlimits_{n to infty} p_{00}(n)=0$? How?
$endgroup$
– Did
Jan 6 at 12:46
$begingroup$
"so state 0 is null" How do you know? Did you prove that $limlimits_{n to infty} p_{00}(n)=0$? How?
$endgroup$
– Did
Jan 6 at 12:46
$begingroup$
I just assumed that if $p_{0,0}=0$, then $lim_{n to infty}p_{0,0}(n)=0$, I know it might be naive, but I'm not aware how to approach this problem.
$endgroup$
– MacAbra
Jan 6 at 13:10
$begingroup$
I just assumed that if $p_{0,0}=0$, then $lim_{n to infty}p_{0,0}(n)=0$, I know it might be naive, but I'm not aware how to approach this problem.
$endgroup$
– MacAbra
Jan 6 at 13:10
$begingroup$
Not especially naive, but squarely wrong because obviously absurd, right? Say, do you have any serious attempt to present?
$endgroup$
– Did
Jan 6 at 13:21
$begingroup$
Not especially naive, but squarely wrong because obviously absurd, right? Say, do you have any serious attempt to present?
$endgroup$
– Did
Jan 6 at 13:21
add a comment |
1 Answer
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$begingroup$
You need to realise that since $p_{0,1} = 1$ and $p_{i,0} + p_{i,i+1} = 1$ then $p_{0,j}$ must be $0$ for all $j ne 1$ and $p_{i,k}$ must be $0$ for all $k ne 0 mbox{ or } i+1$ .
Let $pi_j = frac{e^{-1}}{j!}$ for all $j ge 0$. Then $pi_j$ is the stationary distribution because
begin{eqnarray}
sum_{i=0}^infty pi_i &=& 1 ,\
sum_{i=0}^infty pi_i p_{i,j} &=& frac{1}{j} pi_{j-1} = pi_j mbox{ for $j ge 1$, and}\
sum_{i=0}^infty pi_i p_{i,0}
&=& sum_{i=0}^infty frac{i}{i+1} pi_i \
&=& sum_{i=0}^infty left(1 - frac{1}{(i+1)}right)pi_i\
&=& e^{-1}sum_{i=0}^infty left(frac{1}{i!} - frac{1}{(i+1)!}right)\
&=& pi_0 .
end{eqnarray}
Consequently, all states are positive recurrent—none are transient or null recurrent.
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
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add a comment |
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1 Answer
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$begingroup$
You need to realise that since $p_{0,1} = 1$ and $p_{i,0} + p_{i,i+1} = 1$ then $p_{0,j}$ must be $0$ for all $j ne 1$ and $p_{i,k}$ must be $0$ for all $k ne 0 mbox{ or } i+1$ .
Let $pi_j = frac{e^{-1}}{j!}$ for all $j ge 0$. Then $pi_j$ is the stationary distribution because
begin{eqnarray}
sum_{i=0}^infty pi_i &=& 1 ,\
sum_{i=0}^infty pi_i p_{i,j} &=& frac{1}{j} pi_{j-1} = pi_j mbox{ for $j ge 1$, and}\
sum_{i=0}^infty pi_i p_{i,0}
&=& sum_{i=0}^infty frac{i}{i+1} pi_i \
&=& sum_{i=0}^infty left(1 - frac{1}{(i+1)}right)pi_i\
&=& e^{-1}sum_{i=0}^infty left(frac{1}{i!} - frac{1}{(i+1)!}right)\
&=& pi_0 .
end{eqnarray}
Consequently, all states are positive recurrent—none are transient or null recurrent.
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
$endgroup$
add a comment |
$begingroup$
You need to realise that since $p_{0,1} = 1$ and $p_{i,0} + p_{i,i+1} = 1$ then $p_{0,j}$ must be $0$ for all $j ne 1$ and $p_{i,k}$ must be $0$ for all $k ne 0 mbox{ or } i+1$ .
Let $pi_j = frac{e^{-1}}{j!}$ for all $j ge 0$. Then $pi_j$ is the stationary distribution because
begin{eqnarray}
sum_{i=0}^infty pi_i &=& 1 ,\
sum_{i=0}^infty pi_i p_{i,j} &=& frac{1}{j} pi_{j-1} = pi_j mbox{ for $j ge 1$, and}\
sum_{i=0}^infty pi_i p_{i,0}
&=& sum_{i=0}^infty frac{i}{i+1} pi_i \
&=& sum_{i=0}^infty left(1 - frac{1}{(i+1)}right)pi_i\
&=& e^{-1}sum_{i=0}^infty left(frac{1}{i!} - frac{1}{(i+1)!}right)\
&=& pi_0 .
end{eqnarray}
Consequently, all states are positive recurrent—none are transient or null recurrent.
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
$endgroup$
add a comment |
$begingroup$
You need to realise that since $p_{0,1} = 1$ and $p_{i,0} + p_{i,i+1} = 1$ then $p_{0,j}$ must be $0$ for all $j ne 1$ and $p_{i,k}$ must be $0$ for all $k ne 0 mbox{ or } i+1$ .
Let $pi_j = frac{e^{-1}}{j!}$ for all $j ge 0$. Then $pi_j$ is the stationary distribution because
begin{eqnarray}
sum_{i=0}^infty pi_i &=& 1 ,\
sum_{i=0}^infty pi_i p_{i,j} &=& frac{1}{j} pi_{j-1} = pi_j mbox{ for $j ge 1$, and}\
sum_{i=0}^infty pi_i p_{i,0}
&=& sum_{i=0}^infty frac{i}{i+1} pi_i \
&=& sum_{i=0}^infty left(1 - frac{1}{(i+1)}right)pi_i\
&=& e^{-1}sum_{i=0}^infty left(frac{1}{i!} - frac{1}{(i+1)!}right)\
&=& pi_0 .
end{eqnarray}
Consequently, all states are positive recurrent—none are transient or null recurrent.
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
$endgroup$
You need to realise that since $p_{0,1} = 1$ and $p_{i,0} + p_{i,i+1} = 1$ then $p_{0,j}$ must be $0$ for all $j ne 1$ and $p_{i,k}$ must be $0$ for all $k ne 0 mbox{ or } i+1$ .
Let $pi_j = frac{e^{-1}}{j!}$ for all $j ge 0$. Then $pi_j$ is the stationary distribution because
begin{eqnarray}
sum_{i=0}^infty pi_i &=& 1 ,\
sum_{i=0}^infty pi_i p_{i,j} &=& frac{1}{j} pi_{j-1} = pi_j mbox{ for $j ge 1$, and}\
sum_{i=0}^infty pi_i p_{i,0}
&=& sum_{i=0}^infty frac{i}{i+1} pi_i \
&=& sum_{i=0}^infty left(1 - frac{1}{(i+1)}right)pi_i\
&=& e^{-1}sum_{i=0}^infty left(frac{1}{i!} - frac{1}{(i+1)!}right)\
&=& pi_0 .
end{eqnarray}
Consequently, all states are positive recurrent—none are transient or null recurrent.
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
edited Jan 7 at 7:17
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
answered Jan 7 at 6:29
lonza leggieralonza leggiera
1,14328
1,14328
add a comment |
add a comment |
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$begingroup$
"so state 0 is null" How do you know? Did you prove that $limlimits_{n to infty} p_{00}(n)=0$? How?
$endgroup$
– Did
Jan 6 at 12:46
$begingroup$
I just assumed that if $p_{0,0}=0$, then $lim_{n to infty}p_{0,0}(n)=0$, I know it might be naive, but I'm not aware how to approach this problem.
$endgroup$
– MacAbra
Jan 6 at 13:10
$begingroup$
Not especially naive, but squarely wrong because obviously absurd, right? Say, do you have any serious attempt to present?
$endgroup$
– Did
Jan 6 at 13:21