Linear Dynamics for the state vector of Markov chain functions

The attached file may be somewhat different from the published versionInternational audienceLet $(\vp(X_n))_n$ be a function of a finite-state Markov chain $(X_n)_n$. In this note, we investigate under which conditions the random variable $\vp(X_n)$ 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 $\vp$, 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