# Article

**Keywords:**

Markov chain; finite irreducible Markov chain; sensitivity analysis; stationary probability; transition probability; lumpability

**Summary:**

Sensitivity analysis of irreducible Markov chains considers an original Markov chain with transition probability matix $P$ and modified Markov chain
with transition probability matrix $P$. For their respective stationary probability vectors $\pi, \tilde{\pi}$,
some of the following charactristics are usually studied: $\|\pi - \tilde{\pi}\|_p$ for asymptotical stability [3], $|\pi_i- \tilde{\pi}_i|, \frac{|\pi_i- \tilde{\pi}_i|}{\pi_i}$ for componentwise stability or sensitivity [1]. For functional transition probabilities, $P=P(t)$ and stationary probability vector $\pi(t)$, derivatives are also used for studying sensitivity of some components of stationary distribution with respect to modifications of $P$ [2]. In special cases, modifications of matrix $P$ leave certain stationary probabilities unchanged. This paper studies some special cases which lead to this behavior of stationary probabilities.