# Stationary distribution of a Markov chain with a random transition matrix

Consider a Markov chain $$\{X_t\}$$ on a finite state $$\mathcal{S} = \{1,\dots, S\}$$ space whose transition matrix $$P$$ is populated by elements of the form $$p_{ij} = P(X_{t+1} = j | X_t = i)$$ and we know that the $$S$$ vector $$p_i = (p_{i,1},\dots, p_{i,S})$$ follows a Dirichlet distribution with concentration parameter $$(\alpha_{i1},\dots,\alpha_{iS})$$, and this is true for $$i \in \mathcal{S}$$. Does the notion of an associated stationary distribution make sense in this context? And if so, can it be described in terms of a Dirichlet distribution?

• @Xi'an No, but you only know the distribution of the elements, not the realizations Commented May 6, 2021 at 14:14

The transition matrix $$P$$ is almost surely strongly irreducible, hence produces a Markov chain with a stationary distribution $$\pi$$ associated with the eigenvalue $$\lambda_1=1$$ of $$P$$, i.e., $$\pi P=\pi$$. According to this paper, when $$S$$ grows to infinity, $$\pi$$ converges to the uniform distribution in total variation.
When considering the case $$S=2$$, the stationary distribution associated with$$P=\left(\begin{matrix} 1-a &a\\ b &1-b\end{matrix} \right)$$ is$$\pi=(b\ \ a)\big/(a+b)$$where $$a\sim\mathcal B(\alpha_1,\alpha_2)$$ and $$b\sim\mathcal B(\alpha_2,\alpha_1)$$, which shows $$\pi$$ is not a Dirichlet variate.
• What about with fixed $S$? Commented May 6, 2021 at 16:26
• Right, thanks, I accepted the answer. May I ask about the case before where a new realization was drawn at each $t$? Commented May 6, 2021 at 16:30