Markov chains: long run proportion of a state

Dependencies:

  1. Markov chain

Let $X = [X_0, X_1, \ldots]$ be a markov chain. Let $N_{j,n}$ be the number of times the markov chain is in state $j$ in the first $n$ steps, i.e., \[ N_{j,n} = \sum_{t=0}^{n-1} \begin{cases}1 & \textrm{ if }X_t = j \\ 0 & \textrm{ if }X_t \neq j\end{cases}. \] Define \[ \pi_j = \lim_{n \to \infty} \frac{N_{j,n}}{n}. \] Then $\pi_j$ is called the long-run proportion of state $j$.

Dependency for:

  1. Markov chains: finite sink is positive recurrent
  2. Markov chains: long run proportion is inverse of time to reenter (incomplete)

Info:

Transitive dependencies:

  1. /analysis/topological-space
  2. /sets-and-relations/countable-set
  3. /sets-and-relations/de-morgan-laws
  4. σ-algebra
  5. Generated σ-algebra
  6. Borel algebra
  7. Measurable function
  8. Generators of the real Borel algebra (incomplete)
  9. Measure
  10. σ-algebra is closed under countable intersections
  11. Group
  12. Ring
  13. Semiring
  14. Matrix
  15. Probability
  16. Conditional probability (incomplete)
  17. Random variable
  18. Markov chain