An elementary proof of existence and uniqueness of stationary distributions for irreducible Markov chains
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913896057536512 |
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| author | Schinazi, Rinaldo B. |
| author_facet | Schinazi, Rinaldo B. |
| contents | Consider an $n\times n$ matrix $P$ with the following properties. All entries in $P$ are positive or $0$, the sum of each row is 1 and for all $i$ and $j$ in $\{1,\dots,n\}$ there exists a natural number $k$ such that the $(i,j)$ entry of the matrix $P^k$ is strictly positive. Then, there exists a unique row vector $v$ with only strictly positive entries, whose sum of entries is 1 and such that $vP=P$. We present a proof of this well-known result that uses only basic algebra and the Bolzano-Weierstrass Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13662 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An elementary proof of existence and uniqueness of stationary distributions for irreducible Markov chains Schinazi, Rinaldo B. Probability Consider an $n\times n$ matrix $P$ with the following properties. All entries in $P$ are positive or $0$, the sum of each row is 1 and for all $i$ and $j$ in $\{1,\dots,n\}$ there exists a natural number $k$ such that the $(i,j)$ entry of the matrix $P^k$ is strictly positive. Then, there exists a unique row vector $v$ with only strictly positive entries, whose sum of entries is 1 and such that $vP=P$. We present a proof of this well-known result that uses only basic algebra and the Bolzano-Weierstrass Theorem. |
| title | An elementary proof of existence and uniqueness of stationary distributions for irreducible Markov chains |
| topic | Probability |
| url | https://arxiv.org/abs/2506.13662 |