Classes and Markov Chains












2












$begingroup$



The Markov chain $(Xn; ngeq)$ has state-space $S = (0, 1, 2, . . .)$, with



$p_{i,0} = frac{1}{4}$ and $p_{i,i+1} = frac{3}{4}$ $forall i geq 0$, so that the transition matrix is



P =$begin{pmatrix} frac{1}{4} & frac{3}{4} & 0 & 0 & ...\ frac{1}{4} & 0 & frac{3}{4} & 0 & ... \ frac{1}{4} & 0 & 0 & frac{3}{4} &... \ vdots & vdots&vdots&vdots& ddots end{pmatrix}$



Find the irreducible classes of intercommunicating states. For each class, state:



(a) whether it is transient, positive recurrent or null recurrent (hint - think about the distribution of the return times - say to state 0 - in this case. From there, you can work out whether the states have a finite or infinite expected return time. Can you work out what sort of states you have here?)



(b) its periodicity.




I have tried to approach the first part with a state space diagram and have found that only state 0 and 1 intercommunicate and that the rest of the states do not. But, it is possible to return to every state at some point (say we got to 3, we can then go to state 4, then state 0, 1, 2 and end up at 3 again. So would I put $0,1,2,3,4, ...$ into one class?



I guess I am also not 100% sure about how to classify the states.



How would I show that this is null recurrent or +ve recurrent? I am not sure how I can show that the expected wait time would be $infty$?






stochastic-processes markov-chains markov-process







share|cite|improve this question









$endgroup$

















    2












    $begingroup$



    The Markov chain $(Xn; ngeq)$ has state-space $S = (0, 1, 2, . . .)$, with



    $p_{i,0} = frac{1}{4}$ and $p_{i,i+1} = frac{3}{4}$ $forall i geq 0$, so that the transition matrix is



    P =$begin{pmatrix} frac{1}{4} & frac{3}{4} & 0 & 0 & ...\ frac{1}{4} & 0 & frac{3}{4} & 0 & ... \ frac{1}{4} & 0 & 0 & frac{3}{4} &... \ vdots & vdots&vdots&vdots& ddots end{pmatrix}$



    Find the irreducible classes of intercommunicating states. For each class, state:



    (a) whether it is transient, positive recurrent or null recurrent (hint - think about the distribution of the return times - say to state 0 - in this case. From there, you can work out whether the states have a finite or infinite expected return time. Can you work out what sort of states you have here?)



    (b) its periodicity.




    I have tried to approach the first part with a state space diagram and have found that only state 0 and 1 intercommunicate and that the rest of the states do not. But, it is possible to return to every state at some point (say we got to 3, we can then go to state 4, then state 0, 1, 2 and end up at 3 again. So would I put $0,1,2,3,4, ...$ into one class?



    I guess I am also not 100% sure about how to classify the states.



    How would I show that this is null recurrent or +ve recurrent? I am not sure how I can show that the expected wait time would be $infty$?






    stochastic-processes markov-chains markov-process







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$



      The Markov chain $(Xn; ngeq)$ has state-space $S = (0, 1, 2, . . .)$, with



      $p_{i,0} = frac{1}{4}$ and $p_{i,i+1} = frac{3}{4}$ $forall i geq 0$, so that the transition matrix is



      P =$begin{pmatrix} frac{1}{4} & frac{3}{4} & 0 & 0 & ...\ frac{1}{4} & 0 & frac{3}{4} & 0 & ... \ frac{1}{4} & 0 & 0 & frac{3}{4} &... \ vdots & vdots&vdots&vdots& ddots end{pmatrix}$



      Find the irreducible classes of intercommunicating states. For each class, state:



      (a) whether it is transient, positive recurrent or null recurrent (hint - think about the distribution of the return times - say to state 0 - in this case. From there, you can work out whether the states have a finite or infinite expected return time. Can you work out what sort of states you have here?)



      (b) its periodicity.




      I have tried to approach the first part with a state space diagram and have found that only state 0 and 1 intercommunicate and that the rest of the states do not. But, it is possible to return to every state at some point (say we got to 3, we can then go to state 4, then state 0, 1, 2 and end up at 3 again. So would I put $0,1,2,3,4, ...$ into one class?



      I guess I am also not 100% sure about how to classify the states.



      How would I show that this is null recurrent or +ve recurrent? I am not sure how I can show that the expected wait time would be $infty$?






      stochastic-processes markov-chains markov-process







      share|cite|improve this question









      $endgroup$





      The Markov chain $(Xn; ngeq)$ has state-space $S = (0, 1, 2, . . .)$, with



      $p_{i,0} = frac{1}{4}$ and $p_{i,i+1} = frac{3}{4}$ $forall i geq 0$, so that the transition matrix is



      P =$begin{pmatrix} frac{1}{4} & frac{3}{4} & 0 & 0 & ...\ frac{1}{4} & 0 & frac{3}{4} & 0 & ... \ frac{1}{4} & 0 & 0 & frac{3}{4} &... \ vdots & vdots&vdots&vdots& ddots end{pmatrix}$



      Find the irreducible classes of intercommunicating states. For each class, state:



      (a) whether it is transient, positive recurrent or null recurrent (hint - think about the distribution of the return times - say to state 0 - in this case. From there, you can work out whether the states have a finite or infinite expected return time. Can you work out what sort of states you have here?)



      (b) its periodicity.




      I have tried to approach the first part with a state space diagram and have found that only state 0 and 1 intercommunicate and that the rest of the states do not. But, it is possible to return to every state at some point (say we got to 3, we can then go to state 4, then state 0, 1, 2 and end up at 3 again. So would I put $0,1,2,3,4, ...$ into one class?



      I guess I am also not 100% sure about how to classify the states.



      How would I show that this is null recurrent or +ve recurrent? I am not sure how I can show that the expected wait time would be $infty$?






      stochastic-processes markov-chains markov-process




      stochastic-processes markov-chains markov-process






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 27 at 18:05









      ʎpoqouʎpoqou

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      3581211






















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          $begingroup$

          Suppose $x=0$, then
          $$begin{align}
          mathbb{P}(X_t=0|X_0 = 0) &= sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)mathbb{P}(X_1=0|X_0 = i)\
          &= dfrac{1}{4}sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)\
          &= 1/4,
          end{align}, $$
          so $x=0$ is recurrent since
          $$sum_n mathbb{P}(X_n=0|X_0 = 0) = +infty. $$



          Now, all states communicate, so you can show that all points are recurrent.



          For the periodicity, notice that $1rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 2 and $1rightarrow 0rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 3, so $1$ has period 1 and as this process is irreducible, this chain is aperiodic.






          share|cite|improve this answer









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            $begingroup$

            Suppose $x=0$, then
            $$begin{align}
            mathbb{P}(X_t=0|X_0 = 0) &= sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)mathbb{P}(X_1=0|X_0 = i)\
            &= dfrac{1}{4}sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)\
            &= 1/4,
            end{align}, $$
            so $x=0$ is recurrent since
            $$sum_n mathbb{P}(X_n=0|X_0 = 0) = +infty. $$



            Now, all states communicate, so you can show that all points are recurrent.



            For the periodicity, notice that $1rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 2 and $1rightarrow 0rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 3, so $1$ has period 1 and as this process is irreducible, this chain is aperiodic.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              Suppose $x=0$, then
              $$begin{align}
              mathbb{P}(X_t=0|X_0 = 0) &= sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)mathbb{P}(X_1=0|X_0 = i)\
              &= dfrac{1}{4}sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)\
              &= 1/4,
              end{align}, $$
              so $x=0$ is recurrent since
              $$sum_n mathbb{P}(X_n=0|X_0 = 0) = +infty. $$



              Now, all states communicate, so you can show that all points are recurrent.



              For the periodicity, notice that $1rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 2 and $1rightarrow 0rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 3, so $1$ has period 1 and as this process is irreducible, this chain is aperiodic.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Suppose $x=0$, then
                $$begin{align}
                mathbb{P}(X_t=0|X_0 = 0) &= sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)mathbb{P}(X_1=0|X_0 = i)\
                &= dfrac{1}{4}sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)\
                &= 1/4,
                end{align}, $$
                so $x=0$ is recurrent since
                $$sum_n mathbb{P}(X_n=0|X_0 = 0) = +infty. $$



                Now, all states communicate, so you can show that all points are recurrent.



                For the periodicity, notice that $1rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 2 and $1rightarrow 0rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 3, so $1$ has period 1 and as this process is irreducible, this chain is aperiodic.






                share|cite|improve this answer









                $endgroup$



                Suppose $x=0$, then
                $$begin{align}
                mathbb{P}(X_t=0|X_0 = 0) &= sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)mathbb{P}(X_1=0|X_0 = i)\
                &= dfrac{1}{4}sum_{i=0}^infty mathbb{P}(X_{t-1}=i|X_0 = 0)\
                &= 1/4,
                end{align}, $$
                so $x=0$ is recurrent since
                $$sum_n mathbb{P}(X_n=0|X_0 = 0) = +infty. $$



                Now, all states communicate, so you can show that all points are recurrent.



                For the periodicity, notice that $1rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 2 and $1rightarrow 0rightarrow 0rightarrow 1$ is a path from $1$ to $1$ with positive probability of size 3, so $1$ has period 1 and as this process is irreducible, this chain is aperiodic.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 30 at 3:44









                Flying DogfishFlying Dogfish

                26612




                26612






























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