Linear Dynamics for the state vector of Markov chain functions
Résumé
Let $(\varphi(X_n))_n$ be a function of a finite-state Markov chain $(X_n)_n$. In this note, we investigate under which conditions the random variable $\varphi(X_n)$ have the same distribution as $Y_n$ (for every $n$), where $(Y_n)_n$ is a Markov chain with fixed transition probability matrix. In other words, for a deterministic function $\varphi$, we investigate the conditions under which $(X_n)_n$ is \textit{weakly lumpable for the state vector}. We show that the set of all probability distributions of $X_0$ such that $(X_n)_n$ is weakly lumpable for the state vector can be finitely generated. The connections between our definition of lumpability and usual one's, as the proportional dynamics property, are discussed.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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