Back to Search
Start Over
A Sufficient Condition for a Unique Invariant Distribution of a Higher-Order Markov Chain
- Source :
- Statistics & Probability Letters, vol. 130, Nov. 2017
- Publication Year :
- 2016
-
Abstract
- We derive a sufficient condition for a $k$-th order homogeneous Markov chain $\mathbf{Z}$ with finite alphabet $\mathcal{Z}$ to have a unique invariant distribution on $\mathcal{Z}^k$. Specifically, let $\mathbf{X}$ be a first-order, stationary Markov chain with finite alphabet $\mathcal{X}$ and a single recurrent class, let $g{:}\ \mathcal{X}\to\mathcal{Z}$ be non-injective, and define the (possibly non-Markovian) process $\mathbf{Y}:=g(\mathbf{X})$ (where $g$ is applied coordinate-wise). If $\mathbf{Z}$ is the $k$-th order Markov approximation of $\mathbf{Y}$, its invariant distribution is unique. We generalize this to non-Markovian processes $\mathbf{X}$.<br />Comment: 11 pages, 1 figure
- Subjects :
- Mathematics - Probability
60J10
Subjects
Details
- Database :
- arXiv
- Journal :
- Statistics & Probability Letters, vol. 130, Nov. 2017
- Publication Type :
- Report
- Accession number :
- edsarx.1611.05219
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.spl.2017.07.006