Markov chains: long run proportion of a state

Dependencies:

1. Markov chain

Let $X=[X_0,X_1,\dots ]$ 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., $$N_{j,n}=\sum _{t=0}^{n-1}\{\begin{array}{ll}1& \text{ if }{X}_t=j\\ 0& \text{ if }{X}_t\ne j\end{array}.$$ Define $${\pi }_j=\underset{n\to \mathrm{\infty }}{lim}\frac{N_{j,n}}{n}.$$ 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:

• Depth: 7
• Number of transitive dependencies: 18

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