question about the expected recurrence times to states
0
$begingroup$
- Consider a finite Markov Chain $R$ on the state space ${0,1,2,cdots, N}$ with transition probability matrix $P=|P_{ij}|_{i,j=0}^N$ consisting of three classes ${0,1,2,cdots,N-1}$ and ${N}$ where $0$ and $N$ are absorbing states, both accessible from $k=1,cdots,N-1$ and ${1,2,cdots,N-1}$ is a transient class. Let $k$ be a state satisfying $0<k<N$. We define a auxiliary process $R$ called "the return process" by altering the first and last row of $P$ so that $P_{0k}=P_{Nk}=1$ and leave the other rows unchanged. The return process $R$ is clearly irreducible. Prove that the expected time until absorption $u_k$ with initial state k in the $R$ process equals $1/(pi_0+pi_N)-1$ where $pi_0+pi_N$ is the stationary probability of being in state $0$ or $N$ for the R process.
Hint: Use the relation between stationary probabilities and expected recurrence times or states
stochastic-processes
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
asked Dec 27 '18 at 7:36
GloriaGloria
64
$endgroup$
add a comment |
0
$begingroup$
- Consider a finite Markov Chain $R$ on the state space ${0,1,2,cdots, N}$ with transition probability matrix $P=|P_{ij}|_{i,j=0}^N$ consisting of three classes ${0,1,2,cdots,N-1}$ and ${N}$ where $0$ and $N$ are absorbing states, both accessible from $k=1,cdots,N-1$ and ${1,2,cdots,N-1}$ is a transient class. Let $k$ be a state satisfying $0<k<N$. We define a auxiliary process $R$ called "the return process" by altering the first and last row of $P$ so that $P_{0k}=P_{Nk}=1$ and leave the other rows unchanged. The return process $R$ is clearly irreducible. Prove that the expected time until absorption $u_k$ with initial state k in the $R$ process equals $1/(pi_0+pi_N)-1$ where $pi_0+pi_N$ is the stationary probability of being in state $0$ or $N$ for the R process.
Hint: Use the relation between stationary probabilities and expected recurrence times or states
stochastic-processes
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
asked Dec 27 '18 at 7:36
GloriaGloria
64
$endgroup$
add a comment |
0
0
0
$begingroup$
- Consider a finite Markov Chain $R$ on the state space ${0,1,2,cdots, N}$ with transition probability matrix $P=|P_{ij}|_{i,j=0}^N$ consisting of three classes ${0,1,2,cdots,N-1}$ and ${N}$ where $0$ and $N$ are absorbing states, both accessible from $k=1,cdots,N-1$ and ${1,2,cdots,N-1}$ is a transient class. Let $k$ be a state satisfying $0<k<N$. We define a auxiliary process $R$ called "the return process" by altering the first and last row of $P$ so that $P_{0k}=P_{Nk}=1$ and leave the other rows unchanged. The return process $R$ is clearly irreducible. Prove that the expected time until absorption $u_k$ with initial state k in the $R$ process equals $1/(pi_0+pi_N)-1$ where $pi_0+pi_N$ is the stationary probability of being in state $0$ or $N$ for the R process.
Hint: Use the relation between stationary probabilities and expected recurrence times or states
stochastic-processes
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
asked Dec 27 '18 at 7:36
GloriaGloria
64
$endgroup$
- Consider a finite Markov Chain $R$ on the state space ${0,1,2,cdots, N}$ with transition probability matrix $P=|P_{ij}|_{i,j=0}^N$ consisting of three classes ${0,1,2,cdots,N-1}$ and ${N}$ where $0$ and $N$ are absorbing states, both accessible from $k=1,cdots,N-1$ and ${1,2,cdots,N-1}$ is a transient class. Let $k$ be a state satisfying $0<k<N$. We define a auxiliary process $R$ called "the return process" by altering the first and last row of $P$ so that $P_{0k}=P_{Nk}=1$ and leave the other rows unchanged. The return process $R$ is clearly irreducible. Prove that the expected time until absorption $u_k$ with initial state k in the $R$ process equals $1/(pi_0+pi_N)-1$ where $pi_0+pi_N$ is the stationary probability of being in state $0$ or $N$ for the R process.
Hint: Use the relation between stationary probabilities and expected recurrence times or states
stochastic-processes
stochastic-processes
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
asked Dec 27 '18 at 7:36
GloriaGloria
64
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
asked Dec 27 '18 at 7:36
GloriaGloria
64
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
edited Dec 27 '18 at 7:51
Sauhard Sharma
953318
953318
asked Dec 27 '18 at 7:36
GloriaGloria
64
asked Dec 27 '18 at 7:36
GloriaGloria
64
asked Dec 27 '18 at 7:36
GloriaGloria
64
64
add a comment |
add a comment |
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