Success runs; frog in the well
- Thread starter hello_123
- Start date
D
Deleted member 4993
Guest
Do you know the definition of invariant measure?
Do you know the definition of positive recurrent?
μ is invariant if and only if [MATH]{\displaystyle \mu \left(\varphi _{t}^{-1}(A)\right)=\mu (A)\qquad \forall t\in T,A\in \Sigma .}[/MATH]
A recurrent state is called positive recurrent if the expected amount of time to return to state j given that the chain started in the state j has finite first moment [MATH]E (\tau_{jj} ) < \infty[/MATH]
soo
[MATH]\mu(j)= \sum_{i=0}^{\infty} \mu (i) p_{ij}[/MATH]
[MATH]p_{ij} = p_i \hspace{0.5cm} \textrm{for}\hspace{0.5cm} j=i+1 \hspace{0.5cm} \textrm{and} \hspace{0.5cm} p_{ij} = q_i \hspace{0.5cm} \textrm{for} \hspace{0.5cm} j=0 [/MATH]
[MATH]\mu (0) = \sum_{i=0} \mu (i) q_i[/MATH][MATH]\mu (k) = \sum_{i=0} \mu(k) p_{k-1}[/MATH]
I don't know what now
A recurrent state is called positive recurrent if the expected amount of time to return to state j given that the chain started in the state j has finite first moment [MATH]E (\tau_{jj} ) < \infty[/MATH]
soo
[MATH]\mu(j)= \sum_{i=0}^{\infty} \mu (i) p_{ij}[/MATH]
[MATH]p_{ij} = p_i \hspace{0.5cm} \textrm{for}\hspace{0.5cm} j=i+1 \hspace{0.5cm} \textrm{and} \hspace{0.5cm} p_{ij} = q_i \hspace{0.5cm} \textrm{for} \hspace{0.5cm} j=0 [/MATH]
[MATH]\mu (0) = \sum_{i=0} \mu (i) q_i[/MATH][MATH]\mu (k) = \sum_{i=0} \mu(k) p_{k-1}[/MATH]
I don't know what now